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|
%------------------------------------------------------------------------------
% File : SWW194+1 : TPTP v6.4.0. Released v5.2.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra 436385, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : fta_436385.1000.p [Bla11]
% Status : Theorem
% Rating : 0.33 v6.4.0, 0.42 v6.3.0, 0.33 v6.2.0, 0.36 v6.1.0, 0.47 v6.0.0, 0.35 v5.5.0, 0.59 v5.4.0, 0.61 v5.3.0, 0.63 v5.2.0
% Syntax : Number of formulae : 1199 ( 430 unit)
% Number of atoms : 2810 ( 765 equality)
% Maximal formula depth : 13 ( 5 average)
% Number of connectives : 1803 ( 192 ~; 51 |; 148 &)
% ( 248 <=>;1164 =>; 0 <=; 0 <~>)
% ( 0 ~|; 0 ~&)
% Number of predicates : 71 ( 0 propositional; 1-3 arity)
% Number of functors : 37 ( 12 constant; 0-3 arity)
% Number of variables : 2543 ( 0 sgn;2538 !; 5 ?)
% Maximal term depth : 13 ( 2 average)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-03-01 11:23:48
%------------------------------------------------------------------------------
%----Relevant facts (994)
fof(fact_ext,axiom,(
! [V_g_2,V_f_2] :
( ! [B_x] : hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)
=> V_f_2 = V_g_2 ) )).
fof(fact__096abs_A_I2_A_K_Ax_J_A_060_061_A1_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_x____)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact__096abs_A_I2_A_K_Ay_J_A_060_061_A1_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_y____)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact__0962_A_K_Ax_A_060_061_A1_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_x____),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact__0962_A_K_Ay_A_060_061_A1_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_y____),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact_two__realpow__ge__one,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) )).
fof(fact_four__x__squared,axiom,(
! [V_x] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).
fof(fact_power2__eq__square__number__of,axiom,(
! [V_w,T_b] :
( ( class_Groups_Omonoid__mult(T_b)
& class_Int_Onumber(T_b) )
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_w)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_w)),c_Int_Onumber__class_Onumber__of(T_b,V_w)) ) )).
fof(fact_abs__power2,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).
fof(fact_power2__abs,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).
fof(fact_one__power2,axiom,(
! [T_a] :
( class_Rings_Osemiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,(
! [V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).
fof(fact_power2__eq__square,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,(
! [V_n,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )).
fof(fact__096_N1_A_060_061_A2_A_K_Ax_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OMin),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_x____)) )).
fof(fact_power3__eq__cube,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),V_a) ) )).
fof(fact__096_N1_A_060_061_A2_A_K_Ay_096,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OMin),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_y____)) )).
fof(fact_eq__number__of__Pls__Min,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OPls) != c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OMin) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,(
! [V_q,V_p,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),V_q)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,(
! [V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_x ) )).
fof(fact_nat__numeral__1__eq__1,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )).
fof(fact_Numeral1__eq1__nat,axiom,(
c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OPls)) )).
fof(fact_power__m1__even,axiom,(
! [V_n,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OMin)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,(
! [V_ry,V_rx,V_ly,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_ry)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,(
! [V_ry,V_rx,V_ly,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,(
! [V_ry,V_rx,V_ly,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry))) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,(
! [V_rx,V_ly,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ly) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,(
! [V_rx,V_ly,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,(
! [V_ry,V_rx,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ry) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,(
! [V_ry,V_rx,V_lx,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )).
fof(fact_real__le__antisym,axiom,(
! [V_w,V_z] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z,V_w)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_z)
=> V_z = V_w ) ) )).
fof(fact_real__le__trans,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_j,V_k)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i,V_k) ) ) )).
fof(fact_real__le__linear,axiom,(
! [V_w,V_z] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z,V_w)
| c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_z) ) )).
fof(fact_real__le__refl,axiom,(
! [V_w] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_w) )).
fof(fact_real__mult__assoc,axiom,(
! [V_z3,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z2),V_z3)) )).
fof(fact_real__mult__commute,axiom,(
! [V_w,V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_w) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_w),V_z) )).
fof(fact_power__even__eq,axiom,(
! [V_n,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,(
! [V_q,V_y,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_q) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),V_q)) ) )).
fof(fact_real__mult__1,axiom,(
! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_z) = V_z )).
fof(fact_semiring__norm_I110_J,axiom,(
! [T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oone__class_Oone(T_a) = c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls)) ) )).
fof(fact_xy,axiom,(
c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),v_x____),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),v_y____),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) )).
fof(fact_abs__power__minus__one,axiom,(
! [V_n,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OMin)),V_n)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_le__special_I4_J,axiom,(
! [V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Groups_Oone__class_Oone(T_a))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_xa_2,c_Int_OBit1(c_Int_OPls)) ) ) )).
fof(fact_le__special_I2_J,axiom,(
! [V_ya_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit1(c_Int_OPls),V_ya_2) ) ) )).
fof(fact_csqrt,axiom,(
! [V_z] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Fundamental__Theorem__Algebra__Mirabelle_Ocsqrt(V_z)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_z )).
fof(fact_abs__minus__one,axiom,(
! [T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> c_Groups_Oabs__class_Oabs(T_a,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OMin)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_numeral__1__eq__1,axiom,(
! [T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_mult__numeral__1,axiom,(
! [V_a,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls))),V_a) = V_a ) )).
fof(fact_mult__numeral__1__right,axiom,(
! [V_a,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls))) = V_a ) )).
fof(fact_power__increasing,axiom,(
! [V_a,V_N,V_n,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_N)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)) ) ) ) )).
fof(fact_one__le__power,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) )).
fof(fact_z,axiom,(
v_z = c_Complex_Ocomplex_OComplex(v_x____,v_y____) )).
fof(fact_assms,axiom,(
c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,v_z) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) )).
fof(fact_number__of__is__id,axiom,(
! [V_k] : c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_k) = V_k )).
fof(fact_number__of__add,axiom,(
! [V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,V_w)) = c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Int_Onumber__class_Onumber__of(T_a,V_w)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,(
! [V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,(
! [V_d,V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,V_c,c_Groups_Oplus__class_Oplus(T_a,V_a,V_d)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,(
! [V_d,V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_d) ) )).
fof(fact_add__number__of__eq,axiom,(
! [V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Int_Onumber__class_Onumber__of(T_a,V_w)) = c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,V_w)) ) )).
fof(fact_times__numeral__code_I5_J,axiom,(
! [V_w,V_v] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_v)),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_w)) = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) )).
fof(fact_less__eq__number__of__int__code,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_k_2),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_l_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_add__number__of__left,axiom,(
! [V_z,V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w),V_z)) = c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,V_w)),V_z) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_b) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) )).
fof(fact_zle__antisym,axiom,(
! [V_w,V_z] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z,V_w)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w,V_z)
=> V_z = V_w ) ) )).
fof(fact_zle__trans,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_j,V_k)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_k) ) ) )).
fof(fact_zle__linear,axiom,(
! [V_w,V_z] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z,V_w)
| c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w,V_z) ) )).
fof(fact_zle__refl,axiom,(
! [V_w] : c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w,V_w) )).
fof(fact_rel__simps_I34_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit1(V_k_2),c_Int_OBit1(V_l_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_less__eq__int__code_I16_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit1(V_k1_2),c_Int_OBit1(V_k2_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I19_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,c_Int_OPls) )).
fof(fact_rel__simps_I31_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OBit0(V_l_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_less__eq__int__code_I13_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k1_2),c_Int_OBit0(V_k2_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I24_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OMin,c_Int_OMin) )).
fof(fact_le__nat__number__of,axiom,(
! [V_v_H_2,V_v_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_H_2))
<=> ( ~ c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_v_2,V_v_H_2)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_v_2,c_Int_OPls) ) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,(
! [V_z,V_y,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Oplus__class_Oplus(T_a,V_y,V_z)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)) ) )).
fof(fact_crossproduct__noteq,axiom,(
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
=> ( ( V_a_2 != V_b_2
& V_c_2 != V_d_2 )
<=> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_d_2)) != c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_d_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_c_2)) ) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_c) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,(
! [V_b,V_m,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_m) ) )).
fof(fact_crossproduct__eq,axiom,(
! [V_za_2,V_xa_2,V_ya_2,V_w_2,T_a] :
( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w_2),V_ya_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_za_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_ya_2))
<=> ( V_w_2 = V_xa_2
| V_ya_2 = V_za_2 ) ) ) )).
fof(fact_real__add__left__mono,axiom,(
! [V_z,V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z,V_x),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z,V_y)) ) )).
fof(fact_real__add__mult__distrib,axiom,(
! [V_w,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z1,V_z2)),V_w) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z2),V_w)) )).
fof(fact_right__distrib__number__of,axiom,(
! [V_c,V_b,V_v,T_b] :
( ( class_Int_Onumber(T_b)
& class_Rings_Osemiring(T_b) )
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),c_Groups_Oplus__class_Oplus(T_b,V_b,V_c)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),V_c)) ) )).
fof(fact_left__distrib__number__of,axiom,(
! [V_v,V_b,V_a,T_b] :
( ( class_Int_Onumber(T_b)
& class_Rings_Osemiring(T_b) )
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Groups_Oplus__class_Oplus(T_b,V_a,V_b)),c_Int_Onumber__class_Onumber__of(T_b,V_v)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),c_Int_Onumber__class_Onumber__of(T_b,V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),c_Int_Onumber__class_Onumber__of(T_b,V_v))) ) )).
fof(fact_add__numeral__0__right,axiom,(
! [V_a,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OPls)) = V_a ) )).
fof(fact_add__numeral__0,axiom,(
! [V_a,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OPls),V_a) = V_a ) )).
fof(fact_rel__simps_I22_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,c_Int_OBit1(V_k_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,V_k_2) ) )).
fof(fact_rel__simps_I32_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OBit1(V_l_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_less__eq__int__code_I14_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k1_2),c_Int_OBit1(V_k2_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I21_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,c_Int_OBit0(V_k_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,V_k_2) ) )).
fof(fact_rel__simps_I27_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OPls)
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,c_Int_OPls) ) )).
fof(fact_rel__simps_I30_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit1(V_k_2),c_Int_OMin)
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,c_Int_OMin) ) )).
fof(fact_rel__simps_I26_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OMin,c_Int_OBit1(V_k_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OMin,V_k_2) ) )).
fof(fact_rel__simps_I20_J,axiom,(
~ c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,c_Int_OMin) )).
fof(fact_rel__simps_I23_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OMin,c_Int_OPls) )).
fof(fact_rel__simps_I28_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OMin)
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,c_Int_OMin) ) )).
fof(fact_add__special_I2_J,axiom,(
! [V_w,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w)) = c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OBit1(c_Int_OPls),V_w)) ) )).
fof(fact_add__special_I3_J,axiom,(
! [V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Groups_Oone__class_Oone(T_a)) = c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,c_Int_OBit1(c_Int_OPls))) ) )).
fof(fact_number__of__Bit1,axiom,(
! [V_w,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(V_w)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w)),c_Int_Onumber__class_Onumber__of(T_a,V_w)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,(
! [V_m,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_m,V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))),V_m) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,(
! [V_a,V_m,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))),V_m) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,(
! [V_m,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))),V_m) ) )).
fof(fact_double__number__of__Bit0,axiom,(
! [V_w,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))),c_Int_Onumber__class_Onumber__of(T_a,V_w)) = c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(V_w)) ) )).
fof(fact_number__of__reorient,axiom,(
! [V_xa_2,V_w_2,T_a] :
( class_Int_Onumber(T_a)
=> ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = V_xa_2
<=> V_xa_2 = c_Int_Onumber__class_Onumber__of(T_a,V_w_2) ) ) )).
fof(fact_eq__number__of,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Int_Oring__char__0(T_a) )
=> ( c_Int_Onumber__class_Onumber__of(T_a,V_xa_2) = c_Int_Onumber__class_Onumber__of(T_a,V_ya_2)
<=> V_xa_2 = V_ya_2 ) ) )).
fof(fact_rel__simps_I51_J,axiom,(
! [V_l_2,V_k_2] :
( c_Int_OBit1(V_k_2) = c_Int_OBit1(V_l_2)
<=> V_k_2 = V_l_2 ) )).
fof(fact_power__one__right,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )).
fof(fact_mult__Pls,axiom,(
! [V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OPls),V_w) = c_Int_OPls )).
fof(fact_rel__simps_I48_J,axiom,(
! [V_l_2,V_k_2] :
( c_Int_OBit0(V_k_2) = c_Int_OBit0(V_l_2)
<=> V_k_2 = V_l_2 ) )).
fof(fact_mult__Bit0,axiom,(
! [V_l,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit0(V_k)),V_l) = c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l)) )).
fof(fact_zpower__zpower,axiom,(
! [V_z,V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),V_z) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y),V_z)) )).
fof(fact_mult__2__right,axiom,(
! [V_z,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oplus__class_Oplus(T_a,V_z,V_z) ) )).
fof(fact_mult__2,axiom,(
! [V_z,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = c_Groups_Oplus__class_Oplus(T_a,V_z,V_z) ) )).
fof(fact_one__add__one__is__two,axiom,(
! [T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a)) = c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) ) )).
fof(fact_pos__zmult__eq__1__iff__lemma,axiom,(
! [V_n,V_m] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n) = c_Groups_Oone__class_Oone(tc_Int_Oint)
=> ( V_m = c_Groups_Oone__class_Oone(tc_Int_Oint)
| V_m = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OMin) ) ) )).
fof(fact_zmult__eq__1__iff,axiom,(
! [V_n_2,V_m_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint)
<=> ( ( V_m_2 = c_Groups_Oone__class_Oone(tc_Int_Oint)
& V_n_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) )
| ( V_m_2 = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OMin)
& V_n_2 = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OMin) ) ) ) )).
fof(fact_le__number__of,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_xa_2,V_ya_2) ) ) )).
fof(fact_number__of__mult,axiom,(
! [V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_v)),c_Int_Onumber__class_Onumber__of(T_a,V_w)) ) )).
fof(fact_arith__simps_I32_J,axiom,(
! [V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_v)),c_Int_Onumber__class_Onumber__of(T_a,V_w)) = c_Int_Onumber__class_Onumber__of(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) ) )).
fof(fact_mult__number__of__left,axiom,(
! [V_z,V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w)),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w))),V_z) ) )).
fof(fact_power__commutes,axiom,(
! [V_n,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).
fof(fact_power__mult__distrib,axiom,(
! [V_n,V_b,V_a,T_a] :
( class_Groups_Ocomm__monoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) )).
fof(fact_power__one,axiom,(
! [V_n,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_power2__sum,axiom,(
! [V_y,V_x,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) ) )).
fof(fact_rel__simps_I39_J,axiom,(
! [V_l] : c_Int_OPls != c_Int_OBit1(V_l) )).
fof(fact_rel__simps_I46_J,axiom,(
! [V_k] : c_Int_OBit1(V_k) != c_Int_OPls )).
fof(fact_one__is__num__one,axiom,(
c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OBit1(c_Int_OPls)) )).
fof(fact_rel__simps_I49_J,axiom,(
! [V_l,V_k] : c_Int_OBit0(V_k) != c_Int_OBit1(V_l) )).
fof(fact_rel__simps_I50_J,axiom,(
! [V_l,V_k] : c_Int_OBit1(V_k) != c_Int_OBit0(V_l) )).
fof(fact_Bit0__Pls,axiom,(
c_Int_OBit0(c_Int_OPls) = c_Int_OPls )).
fof(fact_rel__simps_I38_J,axiom,(
! [V_l_2] :
( c_Int_OPls = c_Int_OBit0(V_l_2)
<=> c_Int_OPls = V_l_2 ) )).
fof(fact_rel__simps_I44_J,axiom,(
! [V_k_2] :
( c_Int_OBit0(V_k_2) = c_Int_OPls
<=> V_k_2 = c_Int_OPls ) )).
fof(fact_power__mult,axiom,(
! [V_n,V_m,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n) ) )).
fof(fact_power__abs,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),V_n) ) )).
fof(fact_Bit1__Min,axiom,(
c_Int_OBit1(c_Int_OMin) = c_Int_OMin )).
fof(fact_rel__simps_I43_J,axiom,(
! [V_l_2] :
( c_Int_OMin = c_Int_OBit1(V_l_2)
<=> c_Int_OMin = V_l_2 ) )).
fof(fact_rel__simps_I47_J,axiom,(
! [V_k_2] :
( c_Int_OBit1(V_k_2) = c_Int_OMin
<=> V_k_2 = c_Int_OMin ) )).
fof(fact_rel__simps_I40_J,axiom,(
c_Int_OMin != c_Int_OPls )).
fof(fact_rel__simps_I37_J,axiom,(
c_Int_OPls != c_Int_OMin )).
fof(fact_rel__simps_I42_J,axiom,(
! [V_l] : c_Int_OMin != c_Int_OBit0(V_l) )).
fof(fact_rel__simps_I45_J,axiom,(
! [V_k] : c_Int_OBit0(V_k) != c_Int_OMin )).
fof(fact_real__sum__squared__expand,axiom,(
! [V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) )).
fof(fact__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Az_A_061_AComplex_Ax_Ay_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,(
~ ! [B_x,B_y] : v_z != c_Complex_Ocomplex_OComplex(B_x,B_y) )).
fof(fact_abs__triangle__ineq,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))) ) )).
fof(fact_order__refl,axiom,(
! [V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x) ) )).
fof(fact_power__m1__odd,axiom,(
! [V_n,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OMin)),c_Nat_OSuc(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = c_Int_Onumber__class_Onumber__of(T_a,c_Int_OMin) ) )).
fof(fact_power__power__power,axiom,(
! [T_a] :
( class_Power_Opower(T_a)
=> c_Power_Opower__class_Opower(T_a) = c_Power_Opower_Opower(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Otimes__class_Otimes(T_a)) ) )).
fof(fact_power2__diff,axiom,(
! [V_y,V_x,T_a] :
( class_Int_Onumber__ring(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) ) )).
fof(fact_zero__le__even__power_H,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) ) )).
fof(fact_real__sqrt__ge__abs1,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) )).
fof(fact_real__sqrt__ge__abs2,axiom,(
! [V_x,V_y] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_y),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) )).
fof(fact_sum__power2__ge__zero,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) ) )).
fof(fact_C_I2_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,v_z,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)))) )).
fof(fact_C_I1_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,v_z,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)))) )).
fof(fact_C_I4_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,v_z,c_Complex_Oii))) )).
fof(fact_C_I3_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,v_z,c_Complex_Oii))) )).
fof(fact_int__0__neq__1,axiom,(
c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Groups_Oone__class_Oone(tc_Int_Oint) )).
fof(fact_odd__nonzero,axiom,(
! [V_z] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_z),V_z) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).
fof(fact_number__of__diff,axiom,(
! [V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_v,V_w)) = c_Groups_Ominus__class_Ominus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Int_Onumber__class_Onumber__of(T_a,V_w)) ) )).
fof(fact_zmult__1,axiom,(
! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z) = V_z )).
fof(fact_zmult__1__right,axiom,(
! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),c_Groups_Oone__class_Oone(tc_Int_Oint)) = V_z )).
fof(fact_zmult__commute,axiom,(
! [V_w,V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z) )).
fof(fact_add__eq__if,axiom,(
! [V_n,V_m] :
( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = V_n )
& ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) ) ) )).
fof(fact_zpower__zadd__distrib,axiom,(
! [V_z,V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_z)) )).
fof(fact_zadd__zmult__distrib2,axiom,(
! [V_z2,V_z1,V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,V_z2)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2)) )).
fof(fact_zdiff__zmult__distrib2,axiom,(
! [V_z2,V_z1,V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z1,V_z2)) = c_Groups_Ominus__class_Ominus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2)) )).
fof(fact_power__Suc__0,axiom,(
! [V_n] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_zmult__assoc,axiom,(
! [V_z3,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_z3)) )).
fof(fact_zadd__zmult__distrib,axiom,(
! [V_w,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,V_z2)),V_w) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) )).
fof(fact_zdiff__zmult__distrib,axiom,(
! [V_w,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z1,V_z2)),V_w) = c_Groups_Ominus__class_Ominus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) )).
fof(fact_nat__power__eq__Suc__0__iff,axiom,(
! [V_m_2,V_xa_2] :
( hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_xa_2),V_m_2) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| V_xa_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).
fof(fact_abs__zmult__eq__1,axiom,(
! [V_n,V_m] :
( c_Groups_Oabs__class_Oabs(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n)) = c_Groups_Oone__class_Oone(tc_Int_Oint)
=> c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_m) = c_Groups_Oone__class_Oone(tc_Int_Oint) ) )).
fof(fact_power_Opower_Opower__Suc,axiom,(
! [V_n_2,V_a_2,V_times_2,V_one_2,T_a] : hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),c_Nat_OSuc(V_n_2)) = hAPP(hAPP(V_times_2,V_a_2),hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),V_n_2)) )).
fof(fact_power_Opower_Opower__0,axiom,(
! [V_a_2,V_times_2,V_one_2,T_a] : hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_one_2 )).
fof(fact_complex__diff,axiom,(
! [V_d,V_c,V_b,V_a] : c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_a,V_b),c_Complex_Ocomplex_OComplex(V_c,V_d)) = c_Complex_Ocomplex_OComplex(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_a,V_c),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b,V_d)) )).
fof(fact_zadd__assoc,axiom,(
! [V_z3,V_z2,V_z1] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,V_z2),V_z3) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z2,V_z3)) )).
fof(fact_Complex__eq__0,axiom,(
! [V_b_2,V_a_2] :
( c_Complex_Ocomplex_OComplex(V_a_2,V_b_2) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)
<=> ( V_a_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
& V_b_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_complex__mod__triangle__ineq2,axiom,(
! [V_a,V_b] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_b,V_a)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_b)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_a)) )).
fof(fact_zadd__left__commute,axiom,(
! [V_z,V_y,V_x] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_y,V_z)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_y,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,V_z)) )).
fof(fact_le__iff__diff__le__0,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2),c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_zadd__commute,axiom,(
! [V_w,V_z] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,V_w) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_w,V_z) )).
fof(fact_zadd__0__right,axiom,(
! [V_z] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_z )).
fof(fact_zadd__0,axiom,(
! [V_z] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_z) = V_z )).
fof(fact_complex__zero__def,axiom,(
c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = c_Complex_Ocomplex_OComplex(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) )).
fof(fact_zero__reorient,axiom,(
! [V_xa_2,T_a] :
( class_Groups_Ozero(T_a)
=> ( c_Groups_Ozero__class_Ozero(T_a) = V_xa_2
<=> V_xa_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_diff__0__right,axiom,(
! [V_a,T_a] :
( class_Groups_Ogroup__add(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).
fof(fact_diff__self,axiom,(
! [V_a,T_a] :
( class_Groups_Ogroup__add(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_eq__iff__diff__eq__0,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Groups_Oab__group__add(T_a)
=> ( V_a_2 = V_b_2
<=> c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_right__minus__eq,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Groups_Ogroup__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = V_b_2 ) ) )).
fof(fact_diff__eq__diff__eq,axiom,(
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Groups_Oab__group__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2)
=> ( V_a_2 = V_b_2
<=> V_c_2 = V_d_2 ) ) ) )).
fof(fact_real__sqrt__eq__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_NthRoot_Osqrt(V_xa_2) = c_NthRoot_Osqrt(V_ya_2)
<=> V_xa_2 = V_ya_2 ) )).
fof(fact_real__sqrt__eq__0__iff,axiom,(
! [V_xa_2] :
( c_NthRoot_Osqrt(V_xa_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
<=> V_xa_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_real__sqrt__zero,axiom,(
c_NthRoot_Osqrt(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) )).
fof(fact_real__sqrt__ge__0__iff,axiom,(
! [V_ya_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_ya_2) ) )).
fof(fact_real__sqrt__le__0__iff,axiom,(
! [V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) ) )).
fof(fact_real__sqrt__ge__zero,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)) ) )).
fof(fact_real__sqrt__eq__zero__cancel,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> ( c_NthRoot_Osqrt(V_x) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
=> V_x = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_real__le__eq__diff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_xa_2,V_ya_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) ) )).
fof(fact_diff__Suc__eq__diff__pred,axiom,(
! [V_n,V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) )).
fof(fact_power__0__Suc,axiom,(
! [V_n,T_a] :
( ( class_Power_Opower(T_a)
& class_Rings_Osemiring__0(T_a) )
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Nat_OSuc(V_n)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_Suc__eq__plus1,axiom,(
! [V_n] : c_Nat_OSuc(V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat)) )).
fof(fact_Suc__eq__plus1__left,axiom,(
! [V_n] : c_Nat_OSuc(V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n) )).
fof(fact_nat__one__le__power,axiom,(
! [V_n,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_i)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n)) ) )).
fof(fact_diff__eq__diff__less__eq,axiom,(
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2) ) ) ) )).
fof(fact_power__eq__0__iff,axiom,(
! [V_n_2,V_a_2,T_a] :
( ( class_Power_Opower(T_a)
& class_Rings_Omult__zero(T_a)
& class_Rings_Ono__zero__divisors(T_a)
& class_Rings_Ozero__neq__one(T_a) )
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_a_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).
fof(fact_diff__add__cancel,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Ogroup__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b),V_b) = V_a ) )).
fof(fact_add__diff__cancel,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Ogroup__add(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_b) = V_a ) )).
fof(fact_abs__minus__commute,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)) = c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a)) ) )).
fof(fact_add__0__left,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) )).
fof(fact_add__0,axiom,(
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) )).
fof(fact_double__zero__sym,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_add__0__right,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).
fof(fact_add_Ocomm__neutral,axiom,(
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).
fof(fact_mult__eq__if,axiom,(
! [V_n,V_m] :
( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
& ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n)) ) ) )).
fof(fact_abs__eq__0,axiom,(
! [V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Groups_Oabs__class_Oabs(T_a,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_abs__zero,axiom,(
! [T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_realpow__two__diff,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Ocomm__ring__1(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)) ) )).
fof(fact_complex__mod__triangle__sub,axiom,(
! [V_z,V_w] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_w),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_w,V_z)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_z))) )).
fof(fact_power__inject__base,axiom,(
! [V_b,V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),c_Nat_OSuc(V_n))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> V_a = V_b ) ) ) ) )).
fof(fact_power__le__imp__le__base,axiom,(
! [V_b,V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),c_Nat_OSuc(V_n)))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) ) )).
fof(fact_real__sqrt__le__mono,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_x),c_NthRoot_Osqrt(V_y)) ) )).
fof(fact_real__sqrt__le__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2) ) )).
fof(fact_power__0__left,axiom,(
! [V_n,T_a] :
( ( class_Power_Opower(T_a)
& class_Rings_Osemiring__0(T_a) )
=> ( ( V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a) )
& ( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_power__eq__if,axiom,(
! [V_p,V_m] :
( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )
& ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) ) ) )).
fof(fact_real__sqrt__mult,axiom,(
! [V_y,V_x] : c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)),c_NthRoot_Osqrt(V_y)) )).
fof(fact_real__sqrt__power,axiom,(
! [V_k,V_x] : c_NthRoot_Osqrt(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),V_k)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)),V_k) )).
fof(fact_real__sqrt__eq__1__iff,axiom,(
! [V_xa_2] :
( c_NthRoot_Osqrt(V_xa_2) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal)
<=> V_xa_2 = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) ) )).
fof(fact_real__sqrt__one,axiom,(
c_NthRoot_Osqrt(c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) )).
fof(fact_diff__bin__simps_I1_J,axiom,(
! [V_k] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_k,c_Int_OPls) = V_k )).
fof(fact_diff__bin__simps_I7_J,axiom,(
! [V_l,V_k] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OBit0(V_k),c_Int_OBit0(V_l)) = c_Int_OBit0(c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_Bit1__def,axiom,(
! [V_k] : c_Int_OBit1(V_k) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_k),V_k) )).
fof(fact_add__Pls__right,axiom,(
! [V_k] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,c_Int_OPls) = V_k )).
fof(fact_add__Pls,axiom,(
! [V_k] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OPls,V_k) = V_k )).
fof(fact_add__Bit0__Bit0,axiom,(
! [V_l,V_k] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OBit0(V_k),c_Int_OBit0(V_l)) = c_Int_OBit0(c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_Bit0__def,axiom,(
! [V_k] : c_Int_OBit0(V_k) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_k) )).
fof(fact_zadd__left__mono,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_i),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_j)) ) )).
fof(fact_plus__numeral__code_I9_J,axiom,(
! [V_w,V_v] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_v),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_w)) = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,V_w)) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_double__eq__0__iff,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_add__0__iff,axiom,(
! [V_a_2,V_b_2,T_a] :
( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
=> ( V_b_2 = c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,(
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) )).
fof(fact_field__power__not__zero,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Oring__1__no__zero__divisors(T_a)
=> ( V_a != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) != c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_Pls__def,axiom,(
c_Int_OPls = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).
fof(fact_zero__le__zpower__abs,axiom,(
! [V_n,V_x] : c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_x)),V_n)) )).
fof(fact_abs__triangle__ineq2__sym,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a))) ) )).
fof(fact_abs__triangle__ineq2,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))) ) )).
fof(fact_abs__triangle__ineq3,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))) ) )).
fof(fact_real__mult__left__cancel,axiom,(
! [V_b_2,V_a_2,V_c_2] :
( V_c_2 != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c_2),V_a_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c_2),V_b_2)
<=> V_a_2 = V_b_2 ) ) )).
fof(fact_real__mult__right__cancel,axiom,(
! [V_b_2,V_a_2,V_c_2] :
( V_c_2 != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a_2),V_c_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b_2),V_c_2)
<=> V_a_2 = V_b_2 ) ) )).
fof(fact_real__zero__not__eq__one,axiom,(
c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != c_Groups_Oone__class_Oone(tc_RealDef_Oreal) )).
fof(fact_zero__le__double__add__iff__zero__le__single__add,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) )).
fof(fact_double__add__le__zero__iff__single__add__le__zero,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_add__nonneg__nonneg,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) )).
fof(fact_add__nonneg__eq__0__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_xa_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_ya_2)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_xa_2,V_ya_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_ya_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) ) )).
fof(fact_add__increasing,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) )).
fof(fact_add__increasing2,axiom,(
! [V_a,V_b,V_c,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) )).
fof(fact_add__nonpos__nonpos,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_abs__of__nonneg,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Groups_Oabs__class_Oabs(T_a,V_a) = V_a ) ) )).
fof(fact_abs__le__zero__iff,axiom,(
! [V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_abs__ge__zero,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)) ) )).
fof(fact_complex__add,axiom,(
! [V_d,V_c,V_b,V_a] : c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_a,V_b),c_Complex_Ocomplex_OComplex(V_c,V_d)) = c_Complex_Ocomplex_OComplex(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_c),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_b,V_d)) )).
fof(fact_Suc3__eq__add__3,axiom,(
! [V_n] : c_Nat_OSuc(c_Nat_OSuc(c_Nat_OSuc(V_n))) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OBit1(c_Int_OPls))),V_n) )).
fof(fact_real__sqrt__mult__self__sum__ge__zero,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y),V_y)))) )).
fof(fact_realpow__Suc__le__self,axiom,(
! [V_n,V_r,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_r)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_r,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_r),c_Nat_OSuc(V_n)),V_r) ) ) ) )).
fof(fact_numeral__3__eq__3,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OBit1(c_Int_OPls))) = c_Nat_OSuc(c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) )).
fof(fact_numeral__1__eq__Suc__0,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit1(c_Int_OPls)) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_Complex__eq__number__of,axiom,(
! [V_w_2,V_b_2,V_a_2] :
( c_Complex_Ocomplex_OComplex(V_a_2,V_b_2) = c_Int_Onumber__class_Onumber__of(tc_Complex_Ocomplex,V_w_2)
<=> ( V_a_2 = c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,V_w_2)
& V_b_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_Complex__eq__1,axiom,(
! [V_b_2,V_a_2] :
( c_Complex_Ocomplex_OComplex(V_a_2,V_b_2) = c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)
<=> ( V_a_2 = c_Groups_Oone__class_Oone(tc_RealDef_Oreal)
& V_b_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_complex__one__def,axiom,(
c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = c_Complex_Ocomplex_OComplex(c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) )).
fof(fact_add__2__eq__Suc,axiom,(
! [V_n] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))),V_n) = c_Nat_OSuc(c_Nat_OSuc(V_n)) )).
fof(fact_add__2__eq__Suc_H,axiom,(
! [V_n] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Nat_OSuc(c_Nat_OSuc(V_n)) )).
fof(fact_real__sqrt__ge__one,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)) ) )).
fof(fact_real__sqrt__le__1__iff,axiom,(
! [V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) ) )).
fof(fact_real__sqrt__ge__1__iff,axiom,(
! [V_ya_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_ya_2) ) )).
fof(fact_mult__diff__mult,axiom,(
! [V_b,V_a,V_y,V_x,T_a] :
( class_Rings_Oring(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),V_b)) ) )).
fof(fact_left__diff__distrib__number__of,axiom,(
! [V_v,V_b,V_a,T_b] :
( ( class_Int_Onumber(T_b)
& class_Rings_Oring(T_b) )
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Groups_Ominus__class_Ominus(T_b,V_a,V_b)),c_Int_Onumber__class_Onumber__of(T_b,V_v)) = c_Groups_Ominus__class_Ominus(T_b,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),c_Int_Onumber__class_Onumber__of(T_b,V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),c_Int_Onumber__class_Onumber__of(T_b,V_v))) ) )).
fof(fact_right__diff__distrib__number__of,axiom,(
! [V_c,V_b,V_v,T_b] :
( ( class_Int_Onumber(T_b)
& class_Rings_Oring(T_b) )
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),c_Groups_Ominus__class_Ominus(T_b,V_b,V_c)) = c_Groups_Ominus__class_Ominus(T_b,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),c_Int_Onumber__class_Onumber__of(T_b,V_v)),V_c)) ) )).
fof(fact_add__number__of__diff1,axiom,(
! [V_c,V_w,V_v,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v),c_Groups_Ominus__class_Ominus(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w),V_c)) = c_Groups_Ominus__class_Ominus(T_a,c_Int_Onumber__class_Onumber__of(T_a,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_v,V_w)),V_c) ) )).
fof(fact_abs__diff__triangle__ineq,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d))),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_c)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_d)))) ) )).
fof(fact_abs__triangle__ineq4,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))) ) )).
fof(fact_real__sqrt__abs2,axiom,(
! [V_x] : c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_x)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x) )).
fof(fact_numeral__2__eq__2,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).
fof(fact_semiring__norm_I115_J,axiom,(
c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) )).
fof(fact_diff__bin__simps_I10_J,axiom,(
! [V_l,V_k] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OBit1(V_k),c_Int_OBit1(V_l)) = c_Int_OBit0(c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_diff__bin__simps_I9_J,axiom,(
! [V_l,V_k] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OBit1(V_k),c_Int_OBit0(V_l)) = c_Int_OBit1(c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_diff__bin__simps_I3_J,axiom,(
! [V_l] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OPls,c_Int_OBit0(V_l)) = c_Int_OBit0(c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OPls,V_l)) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,(
! [V_q,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(V_q)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,(
! [V_q,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(V_q)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,(
! [V_q,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) ) )).
fof(fact_power__Suc2,axiom,(
! [V_n,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) ) )).
fof(fact_power__Suc,axiom,(
! [V_n,V_a,T_a] :
( class_Power_Opower(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,(
! [V_q,V_p,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_p,V_q)) ) )).
fof(fact_power__add,axiom,(
! [V_n,V_m,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).
fof(fact_mult__Bit1,axiom,(
! [V_l,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit1(V_k)),V_l) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l)),V_l) )).
fof(fact_add__Bit1__Bit0,axiom,(
! [V_l,V_k] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OBit1(V_k),c_Int_OBit0(V_l)) = c_Int_OBit1(c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_add__Bit0__Bit1,axiom,(
! [V_l,V_k] : c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Int_OBit0(V_k),c_Int_OBit1(V_l)) = c_Int_OBit1(c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_l)) )).
fof(fact_sum__squares__eq__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_xa_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ya_2),V_ya_2)) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_ya_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_add__scale__eq__noteq,axiom,(
! [V_d,V_c,V_b,V_a,V_r,T_a] :
( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
=> ( V_r != c_Groups_Ozero__class_Ozero(T_a)
=> ( ( V_a = V_b
& V_c != V_d )
=> c_Groups_Oplus__class_Oplus(T_a,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)) != c_Groups_Oplus__class_Oplus(T_a,V_b,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_d)) ) ) ) )).
fof(fact_complex__mult,axiom,(
! [V_d,V_c,V_b,V_a] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Ocomplex_OComplex(V_a,V_b)),c_Complex_Ocomplex_OComplex(V_c,V_d)) = c_Complex_Ocomplex_OComplex(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b),V_d)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_d),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b),V_c))) )).
fof(fact_power__mono,axiom,(
! [V_n,V_b,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) ) ) )).
fof(fact_zero__le__power,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) )).
fof(fact_semiring__norm_I112_J,axiom,(
! [T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Groups_Ozero__class_Ozero(T_a) = c_Int_Onumber__class_Onumber__of(T_a,c_Int_OPls) ) )).
fof(fact_number__of__Pls,axiom,(
! [T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Int_OPls) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,(
! [V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_power__0,axiom,(
! [V_a,T_a] :
( class_Power_Opower(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_nat__number__of__Pls,axiom,(
c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OPls) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_semiring__norm_I113_J,axiom,(
c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OPls) )).
fof(fact_zero__is__num__zero,axiom,(
c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OPls) )).
fof(fact_real__two__squares__add__zero__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_xa_2),V_xa_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_ya_2),V_ya_2)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
& V_ya_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_real__sqrt__two__ge__zero,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).
fof(fact_complex__norm,axiom,(
! [V_y,V_x] : c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_x,V_y)) = c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) )).
fof(fact_le__real__sqrt__sumsq,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y),V_y)))) )).
fof(fact_real__sqrt__unique,axiom,(
! [V_x,V_y] :
( hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_x
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y)
=> c_NthRoot_Osqrt(V_x) = V_y ) ) )).
fof(fact_real__sqrt__pow2,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_x ) )).
fof(fact_real__sqrt__pow2__iff,axiom,(
! [V_xa_2] :
( hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_xa_2)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_xa_2
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_xa_2) ) )).
fof(fact_real__squared__diff__one__factored,axiom,(
! [V_x,T_a] :
( class_Rings_Oring__1(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x),c_Groups_Oone__class_Oone(T_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_x,c_Groups_Oone__class_Oone(T_a))),c_Groups_Ominus__class_Ominus(T_a,V_x,c_Groups_Oone__class_Oone(T_a))) ) )).
fof(fact_real__sqrt__sum__squares__eq__cancel2,axiom,(
! [V_y,V_x] :
( c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) = V_y
=> V_x = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_real__sqrt__sum__squares__eq__cancel,axiom,(
! [V_y,V_x] :
( c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) = V_x
=> V_y = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_diff__bin__simps_I4_J,axiom,(
! [V_l] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OPls,c_Int_OBit1(V_l)) = c_Int_OBit1(c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OMin,V_l)) )).
fof(fact_diff__bin__simps_I6_J,axiom,(
! [V_l] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OMin,c_Int_OBit1(V_l)) = c_Int_OBit0(c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OMin,V_l)) )).
fof(fact_diff__bin__simps_I5_J,axiom,(
! [V_l] : c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OMin,c_Int_OBit0(V_l)) = c_Int_OBit1(c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Int_OMin,V_l)) )).
fof(fact_sum__squares__le__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_xa_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ya_2),V_ya_2)),c_Groups_Ozero__class_Ozero(T_a))
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_ya_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_sum__squares__ge__zero,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__ring(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y))) ) )).
fof(fact_linorder__le__cases,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) )).
fof(fact_le__funE,axiom,(
! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
=> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) )).
fof(fact_xt1_I6_J,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x) ) ) ) )).
fof(fact_xt1_I5_J,axiom,(
! [V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> V_x = V_y ) ) ) )).
fof(fact_order__trans,axiom,(
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z) ) ) ) )).
fof(fact_order__antisym,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> V_x = V_y ) ) ) )).
fof(fact_xt1_I4_J,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) )).
fof(fact_ord__le__eq__trans,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) )).
fof(fact_xt1_I3_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) )).
fof(fact_ord__eq__le__trans,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) )).
fof(fact_order__antisym__conv,axiom,(
! [V_xa_2,V_ya_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
<=> V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_le__funD,axiom,(
! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
=> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) )).
fof(fact_order__eq__refl,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( V_x = V_y
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) )).
fof(fact_order__eq__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_xa_2 = V_ya_2
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) ) )).
fof(fact_linorder__linear,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
| c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) )).
fof(fact_le__fun__def,axiom,(
! [V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
<=> ! [B_x] : c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)) ) ) )).
fof(fact_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oab__semigroup__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).
fof(fact_number__of__Bit0,axiom,(
! [V_w,T_a] :
( class_Int_Onumber__ring(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(V_w)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w)),c_Int_Onumber__class_Onumber__of(T_a,V_w)) ) )).
fof(fact_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oab__semigroup__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) )).
fof(fact_add__left__cancel,axiom,(
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2)
<=> V_b_2 = V_c_2 ) ) )).
fof(fact_add__right__cancel,axiom,(
! [V_c_2,V_a_2,V_b_2,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2) = c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2)
<=> V_b_2 = V_c_2 ) ) )).
fof(fact_add__left__imp__eq,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)
=> V_b = V_c ) ) )).
fof(fact_add__imp__eq,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Ocancel__ab__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)
=> V_b = V_c ) ) )).
fof(fact_add__right__imp__eq,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_b,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a)
=> V_b = V_c ) ) )).
fof(fact_zero__le__power__abs,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),V_n)) ) )).
fof(fact_one__reorient,axiom,(
! [V_xa_2,T_a] :
( class_Groups_Oone(T_a)
=> ( c_Groups_Oone__class_Oone(T_a) = V_xa_2
<=> V_xa_2 = c_Groups_Oone__class_Oone(T_a) ) ) )).
fof(fact_eq__number__of__0,axiom,(
! [V_v_2] :
( c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_v_2,c_Int_OPls) ) )).
fof(fact_eq__0__number__of,axiom,(
! [V_v_2] :
( c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_v_2,c_Int_OPls) ) )).
fof(fact_real__sqrt__sum__squares__ge__zero,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) )).
fof(fact_abs__idempotent,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a)) = c_Groups_Oabs__class_Oabs(T_a,V_a) ) )).
fof(fact_odd__0__le__power__imp__0__le,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) ) ) )).
fof(fact_complex_Oinject,axiom,(
! [V_real2_H_2,V_real1_H_2,V_real2_2,V_real1_2] :
( c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2) = c_Complex_Ocomplex_OComplex(V_real1_H_2,V_real2_H_2)
<=> ( V_real1_2 = V_real1_H_2
& V_real2_2 = V_real2_H_2 ) ) )).
fof(fact_real__sqrt__sum__squares__mult__ge__zero,axiom,(
! [V_ya,V_xa,V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_xa),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_ya),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))) )).
fof(fact_le__special_I3_J,axiom,(
! [V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_xa_2,c_Int_OPls) ) ) )).
fof(fact_le__special_I1_J,axiom,(
! [V_ya_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Int_OPls,V_ya_2) ) ) )).
fof(fact_power__decreasing,axiom,(
! [V_a,V_N,V_n,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_N)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) ) )).
fof(fact_nat__mult__2,axiom,(
! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_z,V_z) )).
fof(fact_nat__mult__2__right,axiom,(
! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_z),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_z,V_z) )).
fof(fact_nat__1__add__1,axiom,(
c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) )).
fof(fact_zero__eq__power2,axiom,(
! [V_a_2,T_a] :
( class_Rings_Oring__1__no__zero__divisors(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_zero__power2,axiom,(
! [T_a] :
( class_Rings_Osemiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_real__sqrt__abs,axiom,(
! [V_x] : c_NthRoot_Osqrt(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x) )).
fof(fact_power2__eq__imp__eq,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> V_x = V_y ) ) ) ) )).
fof(fact_power2__le__imp__le,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) ) )).
fof(fact_zero__le__power2,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).
fof(fact_sum__power2__eq__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_xa_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_ya_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_ya_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_real__sqrt__sum__squares__triangle__ineq,axiom,(
! [V_d,V_b,V_c,V_a] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_c)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_b,V_d)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_b),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_c),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_d),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))) )).
fof(fact_real__sqrt__sum__squares__ge2,axiom,(
! [V_x,V_y] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_y,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) )).
fof(fact_real__sqrt__sum__squares__ge1,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) )).
fof(fact_real__sqrt__sum__squares__mult__squared__eq,axiom,(
! [V_ya,V_xa,V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_xa),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_ya),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_xa),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_ya),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) )).
fof(fact_realpow__two__sum__zero__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_xa_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_ya_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
& V_ya_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_add__le__imp__le__left,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) )).
fof(fact_add__le__imp__le__right,axiom,(
! [V_b,V_c,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) )).
fof(fact_add__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) )).
fof(fact_add__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) ) ) )).
fof(fact_add__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ) )).
fof(fact_add__le__cancel__left,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) )).
fof(fact_add__le__cancel__right,axiom,(
! [V_b_2,V_c_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) )).
fof(fact_mult_Ocomm__neutral,axiom,(
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).
fof(fact_mult__1__right,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).
fof(fact_mult__1,axiom,(
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).
fof(fact_mult__1__left,axiom,(
! [V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).
fof(fact_abs__le__D1,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) )).
fof(fact_abs__ge__self,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Oabs__class_Oabs(T_a,V_a)) ) )).
fof(fact_abs__add__abs,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)) ) )).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J,axiom,(
! [V_n,V_x,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n))) ) )).
fof(fact_power__odd__eq,axiom,(
! [V_n,V_a,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).
fof(fact_sum__power2__le__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_xa_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_ya_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(T_a))
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_ya_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_even__power__le__0__imp__0,axiom,(
! [V_k,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_k)),c_Groups_Ozero__class_Ozero(T_a))
=> V_a = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).
fof(fact_norm__diff__triangle__ineq,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_c)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_d)))) ) )).
fof(fact_convex__bound__le,axiom,(
! [V_v,V_u,V_y,V_a,V_x,T_a] :
( class_Rings_Olinordered__semiring__1(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_u)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_v)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_u,V_v) = c_Groups_Oone__class_Oone(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y)),V_a) ) ) ) ) ) ) )).
fof(fact_norm__number__of,axiom,(
! [V_w,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_RealVector_Oreal__normed__algebra__1(T_a) )
=> c_RealVector_Onorm__class_Onorm(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,V_w)) ) )).
fof(fact_norm__triangle__ineq3,axiom,(
! [V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b))),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))) ) )).
fof(fact_norm__power__ineq,axiom,(
! [V_n,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra__1(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_n)) ) )).
fof(fact_norm__triangle__ineq4,axiom,(
! [V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b))) ) )).
fof(fact_complex__i__not__one,axiom,(
c_Complex_Oii != c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) )).
fof(fact_complex__i__not__number__of,axiom,(
! [V_w] : c_Complex_Oii != c_Int_Onumber__class_Onumber__of(tc_Complex_Ocomplex,V_w) )).
fof(fact_complex__i__not__zero,axiom,(
c_Complex_Oii != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) )).
fof(fact_real__norm__def,axiom,(
! [V_r] : c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,V_r) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_r) )).
fof(fact_i__squared,axiom,(
hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Oii),c_Complex_Oii) = c_Int_Onumber__class_Onumber__of(tc_Complex_Ocomplex,c_Int_OMin) )).
fof(fact_Complex__eq__i,axiom,(
! [V_ya_2,V_xa_2] :
( c_Complex_Ocomplex_OComplex(V_xa_2,V_ya_2) = c_Complex_Oii
<=> ( V_xa_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
& V_ya_2 = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) ) ) )).
fof(fact_i__def,axiom,(
c_Complex_Oii = c_Complex_Ocomplex_OComplex(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )).
fof(fact_mult__zero__left,axiom,(
! [V_a,T_a] :
( class_Rings_Omult__zero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult_Ozero__left,axiom,(
! [V_b,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult__left_Ozero,axiom,(
! [V_y,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult__zero__right,axiom,(
! [V_a,T_a] :
( class_Rings_Omult__zero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult_Ozero__right,axiom,(
! [V_a,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult__right_Ozero,axiom,(
! [V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_mult__eq__0__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Rings_Oring__no__zero__divisors(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_a_2 = c_Groups_Ozero__class_Ozero(T_a)
| V_b_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_no__zero__divisors,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Ono__zero__divisors(T_a)
=> ( V_a != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_b != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) != c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_divisors__zero,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Ono__zero__divisors(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a)
=> ( V_a = c_Groups_Ozero__class_Ozero(T_a)
| V_b = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_one__neq__zero,axiom,(
! [T_a] :
( class_Rings_Ozero__neq__one(T_a)
=> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_zero__neq__one,axiom,(
! [T_a] :
( class_Rings_Ozero__neq__one(T_a)
=> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_mult_Oadd__right,axiom,(
! [V_b_H,V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oplus__class_Oplus(T_a,V_b,V_b_H)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b_H)) ) )).
fof(fact_mult__right_Oadd,axiom,(
! [V_y,V_x,V_xa,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_y)) ) )).
fof(fact_comm__semiring__class_Odistrib,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_c) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).
fof(fact_mult_Oadd__left,axiom,(
! [V_b,V_a_H,V_a,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_a_H)),V_b) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b)) ) )).
fof(fact_mult__left_Oadd,axiom,(
! [V_ya,V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),V_ya) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_ya),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_ya)) ) )).
fof(fact_combine__common__factor,axiom,(
! [V_c,V_b,V_e,V_a,T_a] :
( class_Rings_Osemiring(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_e),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_e),V_c)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_e),V_c) ) )).
fof(fact_mult_Odiff__right,axiom,(
! [V_b_H,V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ominus__class_Ominus(T_a,V_b,V_b_H)) = c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b_H)) ) )).
fof(fact_mult__right_Odiff,axiom,(
! [V_y,V_x,V_xa,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)) = c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_y)) ) )).
fof(fact_mult_Odiff__left,axiom,(
! [V_b,V_a_H,V_a,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a,V_a_H)),V_b) = c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b)) ) )).
fof(fact_mult__left_Odiff,axiom,(
! [V_ya,V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)),V_ya) = c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_ya),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_ya)) ) )).
fof(fact_norm__eq__zero,axiom,(
! [V_xa_2,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ( c_RealVector_Onorm__class_Onorm(T_a,V_xa_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)
<=> V_xa_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_norm__zero,axiom,(
! [T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_abs__mult,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)) ) )).
fof(fact_abs__mult__self,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) ) )).
fof(fact_abs__one,axiom,(
! [T_a] :
( class_Rings_Olinordered__idom(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Oone__class_Oone(T_a) ) )).
fof(fact_norm__minus__commute,axiom,(
! [V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)) = c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a)) ) )).
fof(fact_norm__ge__zero,axiom,(
! [V_x,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)) ) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,V_y)) ) ) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)) ) ) )).
fof(fact_abs__norm__cancel,axiom,(
! [V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a)) = c_RealVector_Onorm__class_Onorm(T_a,V_a) ) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,(
! [V_n,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x)
=> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)) ) )).
fof(fact_power2__i,axiom,(
hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Complex_Oii),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Int_Onumber__class_Onumber__of(tc_Complex_Ocomplex,c_Int_OMin) )).
fof(fact_zero__le__square,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__ring(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)) ) )).
fof(fact_zero__le__mult__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2))
<=> ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) )).
fof(fact_mult__le__0__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2),c_Groups_Ozero__class_Ozero(T_a))
<=> ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) ) ) ) ) )).
fof(fact_mult__nonneg__nonneg,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__cancel__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) ) ) ) )).
fof(fact_mult__nonneg__nonpos,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__cancel__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__nonneg__nonpos2,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__cancel__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__nonpos__nonneg,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__cancel__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__nonpos__nonpos,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) ) ) ) )).
fof(fact_mult__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Oordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) ) ) )).
fof(fact_mult__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Oordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_comm__mult__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Oordered__comm__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_mult__right__mono__neg,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) ) ) )).
fof(fact_mult__left__mono__neg,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_mult__mono_H,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Oordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_mult__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Oordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_split__mult__pos__le,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) ) )
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) ) ) )).
fof(fact_split__mult__neg__le,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__cancel__semiring(T_a)
=> ( ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) ) )
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_not__one__le__zero,axiom,(
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ~ c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_zero__le__one,axiom,(
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)) ) )).
fof(fact_mult_Oprod__diff__prod,axiom,(
! [V_b,V_a,V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b))) ) )).
fof(fact_eq__add__iff2,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oring(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2)
<=> V_c_2 = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2) ) ) )).
fof(fact_eq__add__iff1,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oring(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2)
<=> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2) = V_d_2 ) ) )).
fof(fact_power__eq__0__iff__number__of,axiom,(
! [V_w_2,V_a_2,T_a] :
( ( class_Power_Opower(T_a)
& class_Rings_Omult__zero(T_a)
& class_Rings_Ono__zero__divisors(T_a)
& class_Rings_Ozero__neq__one(T_a) )
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w_2)) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_a_2 = c_Groups_Ozero__class_Ozero(T_a)
& c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w_2) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).
fof(fact_norm__le__zero__iff,axiom,(
! [V_xa_2,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_xa_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
<=> V_xa_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_norm__diff__ineq,axiom,(
! [V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b))) ) )).
fof(fact_norm__triangle__ineq2,axiom,(
! [V_b,V_a,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))) ) )).
fof(fact_norm__mult,axiom,(
! [V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__div__algebra(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_RealVector_Onorm__class_Onorm(T_a,V_y)) ) )).
fof(fact_norm__one,axiom,(
! [T_a] :
( class_RealVector_Oreal__normed__algebra__1(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) ) )).
fof(fact_norm__power,axiom,(
! [V_n,V_x,T_a] :
( class_RealVector_Oreal__normed__div__algebra(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_n) ) )).
fof(fact_mult__right__le__one__le,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y),V_x) ) ) ) ) )).
fof(fact_mult__left__le__one__le,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x),V_x) ) ) ) ) )).
fof(fact_le__add__iff2,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2)) ) ) )).
fof(fact_le__add__iff1,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2),V_d_2) ) ) )).
fof(fact_abs__mult__pos,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_y)),V_x) = c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)) ) ) )).
fof(fact_abs__eq__mult,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Oordered__ring__abs(T_a)
=> ( ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
| c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) )
& ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
| c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) ) )
=> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)) ) ) )).
fof(fact_power__0__left__number__of,axiom,(
! [V_w,T_a] :
( ( class_Power_Opower(T_a)
& class_Rings_Osemiring__0(T_a) )
=> ( ( c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w)) = c_Groups_Oone__class_Oone(T_a) )
& ( c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_w)) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_norm__mult__ineq,axiom,(
! [V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_RealVector_Onorm__class_Onorm(T_a,V_y))) ) )).
fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) )).
fof(fact_norm__triangle__ineq,axiom,(
! [V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),c_RealVector_Onorm__class_Onorm(T_a,V_y))) ) )).
fof(fact_realpow__num__eq__if,axiom,(
! [V_m,V_n,T_a] :
( class_Power_Opower(T_a)
=> ( ( V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),V_n) = c_Groups_Oone__class_Oone(T_a) )
& ( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) ) ) ) )).
fof(fact_number__of1,axiom,(
! [V_n] :
( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_n))
=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OBit0(V_n)))
& c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OBit1(V_n))) ) ) )).
fof(fact_lemma__realpow__diff,axiom,(
! [V_y,V_n,V_p,T_a] :
( class_Groups_Omonoid__mult(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_p,V_n)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_n),V_p)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_p))),V_y) ) ) )).
fof(fact_le0,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) )).
fof(fact_norm__ratiotest__lemma,axiom,(
! [V_y,V_x,V_c,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_c,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c),c_RealVector_Onorm__class_Onorm(T_a,V_y)))
=> V_x = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_n__not__Suc__n,axiom,(
! [V_n] : V_n != c_Nat_OSuc(V_n) )).
fof(fact_Suc__n__not__n,axiom,(
! [V_n] : c_Nat_OSuc(V_n) != V_n )).
fof(fact_nat_Oinject,axiom,(
! [V_nat_H_2,V_nat_2] :
( c_Nat_OSuc(V_nat_2) = c_Nat_OSuc(V_nat_H_2)
<=> V_nat_2 = V_nat_H_2 ) )).
fof(fact_Suc__inject,axiom,(
! [V_y,V_x] :
( c_Nat_OSuc(V_x) = c_Nat_OSuc(V_y)
=> V_x = V_y ) )).
fof(fact_nat__add__commute,axiom,(
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m) )).
fof(fact_nat__add__left__commute,axiom,(
! [V_z,V_y,V_x] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,V_z)) )).
fof(fact_nat__add__assoc,axiom,(
! [V_k,V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) )).
fof(fact_nat__add__left__cancel,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)
<=> V_m_2 = V_n_2 ) )).
fof(fact_nat__add__right__cancel,axiom,(
! [V_n_2,V_k_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_k_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_k_2)
<=> V_m_2 = V_n_2 ) )).
fof(fact_le__refl,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_n) )).
fof(fact_nat__le__linear,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) ) )).
fof(fact_eq__imp__le,axiom,(
! [V_n,V_m] :
( V_m = V_n
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_le__trans,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_k)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_k) ) ) )).
fof(fact_le__antisym,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> V_m = V_n ) ) )).
fof(fact_nat__mult__commute,axiom,(
! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m) )).
fof(fact_nat__mult__assoc,axiom,(
! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).
fof(fact_diff__commute,axiom,(
! [V_k,V_j,V_i] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_k),V_j) )).
fof(fact_Zero__not__Suc,axiom,(
! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_m) )).
fof(fact_nat_Osimps_I2_J,axiom,(
! [V_nat_H] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_nat_H) )).
fof(fact_Suc__not__Zero,axiom,(
! [V_m] : c_Nat_OSuc(V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_nat_Osimps_I3_J,axiom,(
! [V_nat_H_1] : c_Nat_OSuc(V_nat_H_1) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_Zero__neq__Suc,axiom,(
! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_m) )).
fof(fact_Suc__neq__Zero,axiom,(
! [V_m] : c_Nat_OSuc(V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_plus__nat_Oadd__0,axiom,(
! [V_n] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) = V_n )).
fof(fact_Nat_Oadd__0__right,axiom,(
! [V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m )).
fof(fact_add__is__0,axiom,(
! [V_n_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).
fof(fact_add__eq__self__zero,axiom,(
! [V_n,V_m] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = V_m
=> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).
fof(fact_less__eq__nat_Osimps_I1_J,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) )).
fof(fact_le__0__eq,axiom,(
! [V_n_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).
fof(fact_add__Suc__shift,axiom,(
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) )).
fof(fact_add__Suc,axiom,(
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) )).
fof(fact_add__Suc__right,axiom,(
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) )).
fof(fact_Suc__n__not__le__n,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n),V_n) )).
fof(fact_not__less__eq__eq,axiom,(
! [V_n_2,V_m_2] :
( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),V_m_2) ) )).
fof(fact_le__Suc__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2))
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
| V_m_2 = c_Nat_OSuc(V_n_2) ) ) )).
fof(fact_Suc__le__mono,axiom,(
! [V_m_2,V_n_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),c_Nat_OSuc(V_m_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2) ) )).
fof(fact_le__SucI,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) )).
fof(fact_le__SucE,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n))
=> ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> V_m = c_Nat_OSuc(V_n) ) ) )).
fof(fact_Suc__leD,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_mult__0,axiom,(
! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_mult__0__right,axiom,(
! [V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_mult__is__0,axiom,(
! [V_n_2,V_m_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).
fof(fact_mult__cancel1,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)
<=> ( V_m_2 = V_n_2
| V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).
fof(fact_mult__cancel2,axiom,(
! [V_n_2,V_k_2,V_m_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2)
<=> ( V_m_2 = V_n_2
| V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).
fof(fact_diffs0__imp__equal,axiom,(
! [V_n,V_m] :
( c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> ( c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> V_m = V_n ) ) )).
fof(fact_diff__self__eq__0,axiom,(
! [V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_minus__nat_Odiff__0,axiom,(
! [V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m )).
fof(fact_diff__0__eq__0,axiom,(
! [V_n] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_Suc__mult__cancel1,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_n_2)
<=> V_m_2 = V_n_2 ) )).
fof(fact_le__add2,axiom,(
! [V_m,V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) )).
fof(fact_le__add1,axiom,(
! [V_m,V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m)) )).
fof(fact_le__Suc__ex__iff,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_l_2)
<=> ? [B_n] : V_l_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,B_n) ) )).
fof(fact_le__iff__add,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
<=> ? [B_k] : V_n_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k) ) )).
fof(fact_nat__add__left__cancel__le,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_trans__le__add1,axiom,(
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)) ) )).
fof(fact_trans__le__add2,axiom,(
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)) ) )).
fof(fact_add__le__mono1,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)) ) )).
fof(fact_add__le__mono,axiom,(
! [V_l,V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_l)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l)) ) ) )).
fof(fact_add__leD2,axiom,(
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) ) )).
fof(fact_add__leD1,axiom,(
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_add__leE,axiom,(
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> ~ ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) ) ) )).
fof(fact_diff__Suc__Suc,axiom,(
! [V_n,V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) )).
fof(fact_Suc__diff__diff,axiom,(
! [V_k,V_n,V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n),c_Nat_OSuc(V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_k) )).
fof(fact_add__mult__distrib2,axiom,(
! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) )).
fof(fact_add__mult__distrib,axiom,(
! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).
fof(fact_le__square,axiom,(
! [V_m] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m)) )).
fof(fact_le__cube,axiom,(
! [V_m] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m))) )).
fof(fact_mult__le__mono1,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k)) ) )).
fof(fact_mult__le__mono2,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j)) ) )).
fof(fact_mult__le__mono,axiom,(
! [V_l,V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_l)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_l)) ) ) )).
fof(fact_diff__add__inverse2,axiom,(
! [V_n,V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_n) = V_m )).
fof(fact_diff__add__inverse,axiom,(
! [V_m,V_n] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m),V_n) = V_m )).
fof(fact_diff__diff__left,axiom,(
! [V_k,V_j,V_i] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)) )).
fof(fact_diff__cancel,axiom,(
! [V_n,V_m,V_k] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) )).
fof(fact_diff__cancel2,axiom,(
! [V_n,V_k,V_m] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) )).
fof(fact_diff__le__self,axiom,(
! [V_n,V_m] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_m) )).
fof(fact_diff__le__mono2,axiom,(
! [V_l,V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_n),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_m)) ) )).
fof(fact_diff__le__mono,axiom,(
! [V_l,V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_l),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_l)) ) )).
fof(fact_diff__diff__cancel,axiom,(
! [V_n,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_n)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_i)) = V_i ) )).
fof(fact_eq__diff__iff,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_m_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_n_2)
=> ( c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_k_2) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_k_2)
<=> V_m_2 = V_n_2 ) ) ) )).
fof(fact_Nat_Odiff__diff__eq,axiom,(
! [V_n,V_m,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_m)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_k),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) ) ) )).
fof(fact_le__diff__iff,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_m_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_n_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_k_2),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_k_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) ) ) )).
fof(fact_diff__mult__distrib2,axiom,(
! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) )).
fof(fact_diff__mult__distrib,axiom,(
! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)),V_k) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).
fof(fact_nat__mult__1,axiom,(
! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = V_n )).
fof(fact_nat__1__eq__mult__iff,axiom,(
! [V_n_2,V_m_2] :
( c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2)
<=> ( V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat)
& V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) ) )).
fof(fact_nat__mult__1__right,axiom,(
! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n )).
fof(fact_nat__mult__eq__1__iff,axiom,(
! [V_n_2,V_m_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
<=> ( V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat)
& V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) ) )).
fof(fact_add__is__1,axiom,(
! [V_n_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> ( ( V_m_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
| ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) )).
fof(fact_one__is__add,axiom,(
! [V_n_2,V_m_2] :
( c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( ( V_m_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
| ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) )).
fof(fact_mult__eq__1__iff,axiom,(
! [V_n_2,V_m_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> ( V_m_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& V_n_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).
fof(fact_One__nat__def,axiom,(
c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_diff__add__0,axiom,(
! [V_m,V_n] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_diff__is__0__eq_H,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).
fof(fact_diff__is__0__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_mult__Suc__right,axiom,(
! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Nat_OSuc(V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) )).
fof(fact_mult__Suc,axiom,(
! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_m)),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) )).
fof(fact_Suc__mult__le__cancel1,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_Suc__diff__le,axiom,(
! [V_m,V_n] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) = c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) ) )).
fof(fact_mult__eq__self__implies__10,axiom,(
! [V_n,V_m] :
( V_m = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)
=> ( V_n = c_Groups_Oone__class_Oone(tc_Nat_Onat)
| V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).
fof(fact_diff__add__assoc2,axiom,(
! [V_i,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_i),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k),V_i) ) )).
fof(fact_add__diff__assoc2,axiom,(
! [V_i,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k),V_i) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_i),V_k) ) )).
fof(fact_diff__add__assoc,axiom,(
! [V_i,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)) ) )).
fof(fact_le__imp__diff__is__add,axiom,(
! [V_k_2,V_j_2,V_i_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2)
=> ( c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2) = V_k_2
<=> V_j_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_i_2) ) ) )).
fof(fact_le__add__diff__inverse2,axiom,(
! [V_m,V_n] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_n) = V_m ) )).
fof(fact_le__diff__conv2,axiom,(
! [V_i_2,V_j_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_j_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_k_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i_2,V_k_2),V_j_2) ) ) )).
fof(fact_add__diff__assoc,axiom,(
! [V_i,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) ) )).
fof(fact_le__add__diff__inverse,axiom,(
! [V_m,V_n] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = V_m ) )).
fof(fact_le__add__diff,axiom,(
! [V_m,V_n,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m),V_k)) ) )).
fof(fact_le__diff__conv,axiom,(
! [V_i_2,V_k_2,V_j_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_k_2),V_i_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i_2,V_k_2)) ) )).
fof(fact_diff__diff__right,axiom,(
! [V_i,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),V_j) ) )).
fof(fact_diff__Suc__1,axiom,(
! [V_n] : c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n )).
fof(fact_one__le__mult__iff,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)
& c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2) ) ) )).
fof(fact_number__of2,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,c_Int_OPls)) )).
fof(fact_diff__Suc__diff__eq2,axiom,(
! [V_m,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)),V_m) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_j),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_m)) ) )).
fof(fact_diff__Suc__diff__eq1,axiom,(
! [V_m,V_j,V_k] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k))) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),c_Nat_OSuc(V_j)) ) )).
fof(fact_rabs__ratiotest__lemma,axiom,(
! [V_y,V_x,V_c] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_c,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_y)))
=> V_x = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) ) )).
fof(fact_complex_Osize_I1_J,axiom,(
! [V_real2,V_real1] : c_Complex_Ocomplex_Ocomplex__size(c_Complex_Ocomplex_OComplex(V_real1,V_real2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_nat__le__add__iff1,axiom,(
! [V_n_2,V_m_2,V_u_2,V_i_2,V_j_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,V_i_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i_2,V_j_2)),V_u_2),V_m_2),V_n_2) ) ) )).
fof(fact_nat__diff__add__eq1,axiom,(
! [V_n,V_m,V_u,V_i,V_j] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_i)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u),V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j)),V_u),V_m),V_n) ) )).
fof(fact_nat__eq__add__iff1,axiom,(
! [V_n_2,V_m_2,V_u_2,V_i_2,V_j_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,V_i_2)
=> ( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2)
<=> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i_2,V_j_2)),V_u_2),V_m_2) = V_n_2 ) ) )).
fof(fact_nat__mult__eq__cancel__disj,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)
<=> ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| V_m_2 = V_n_2 ) ) )).
fof(fact_left__add__mult__distrib,axiom,(
! [V_k,V_j,V_u,V_i] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u),V_k)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j)),V_u),V_k) )).
fof(fact_nat__eq__add__iff2,axiom,(
! [V_n_2,V_m_2,V_u_2,V_j_2,V_i_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2)
=> ( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2)
<=> V_m_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2)),V_u_2),V_n_2) ) ) )).
fof(fact_nat__diff__add__eq2,axiom,(
! [V_n,V_m,V_u,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u),V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_i)),V_u),V_n)) ) )).
fof(fact_nat__le__add__iff2,axiom,(
! [V_n_2,V_m_2,V_u_2,V_j_2,V_i_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2)),V_u_2),V_n_2)) ) ) )).
fof(fact_cos__x__y__le__one,axiom,(
! [V_y,V_x] : c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_x,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) )).
fof(fact_complex_Osize_I2_J,axiom,(
! [V_real2,V_real1] : c_Nat_Osize__class_Osize(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_real1,V_real2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).
fof(fact_complex_Osimps_I2_J,axiom,(
! [V_real2_2,V_real1_2,V_f1_2,T_a] : c_Complex_Ocomplex_Ocomplex__case(T_a,V_f1_2,c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2)) = hAPP(hAPP(V_f1_2,V_real1_2),V_real2_2) )).
fof(fact_divide__zero__left,axiom,(
! [V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_divide_Ozero,axiom,(
! [V_y,T_a] :
( class_RealVector_Oreal__normed__field(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_divide__zero,axiom,(
! [V_a,T_a] :
( class_Rings_Odivision__ring__inverse__zero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_diff__divide__distrib,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b),V_c) = c_Groups_Ominus__class_Ominus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) ) )).
fof(fact_divide_Odiff,axiom,(
! [V_ya,V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__field(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,V_x,V_y),V_ya) = c_Groups_Ominus__class_Ominus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_ya),c_Rings_Oinverse__class_Odivide(T_a,V_y,V_ya)) ) )).
fof(fact_times__divide__eq__right,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) = c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) ) )).
fof(fact_add__divide__distrib,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) ) )).
fof(fact_divide_Oadd,axiom,(
! [V_ya,V_y,V_x,T_a] :
( class_RealVector_Oreal__normed__field(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y),V_ya) = c_Groups_Oplus__class_Oplus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_ya),c_Rings_Oinverse__class_Odivide(T_a,V_y,V_ya)) ) )).
fof(fact_divide__1,axiom,(
! [V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) = V_a ) )).
fof(fact_power__divide,axiom,(
! [V_n,V_b,V_a,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)),V_n) = c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) )).
fof(fact_real__divide__square__eq,axiom,(
! [V_a,V_r] : c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_r)) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_a,V_r) )).
fof(fact_real__sqrt__divide,axiom,(
! [V_y,V_x] : c_NthRoot_Osqrt(c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_x,V_y)) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_x),c_NthRoot_Osqrt(V_y)) )).
fof(fact_nonzero__norm__divide,axiom,(
! [V_a,V_b,T_a] :
( class_RealVector_Oreal__normed__field(T_a)
=> ( V_b != c_Groups_Ozero__class_Ozero(T_a)
=> c_RealVector_Onorm__class_Onorm(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)) ) ) )).
fof(fact_norm__divide,axiom,(
! [V_b,V_a,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_RealVector_Oreal__normed__field(T_a) )
=> c_RealVector_Onorm__class_Onorm(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)) ) )).
fof(fact_eq__divide__imp,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_c != c_Groups_Ozero__class_Ozero(T_a)
=> ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b
=> V_a = c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c) ) ) ) )).
fof(fact_divide__eq__imp,axiom,(
! [V_a,V_b,V_c,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_c != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)
=> c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c) = V_a ) ) ) )).
fof(fact_nonzero__divide__eq__eq,axiom,(
! [V_a_2,V_b_2,V_c_2,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2) = V_a_2
<=> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2) ) ) ) )).
fof(fact_nonzero__eq__divide__eq,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_a_2 = c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2)
<=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2) = V_b_2 ) ) ) )).
fof(fact_divide__self__if,axiom,(
! [V_a,T_a] :
( class_Rings_Odivision__ring__inverse__zero(T_a)
=> ( ( V_a = c_Groups_Ozero__class_Ozero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a) )
& ( V_a != c_Groups_Ozero__class_Ozero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,V_a) = c_Groups_Oone__class_Oone(T_a) ) ) ) )).
fof(fact_divide__self,axiom,(
! [V_a,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_a != c_Groups_Ozero__class_Ozero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,V_a) = c_Groups_Oone__class_Oone(T_a) ) ) )).
fof(fact_right__inverse__eq,axiom,(
! [V_a_2,V_b_2,T_a] :
( class_Rings_Odivision__ring(T_a)
=> ( V_b_2 != c_Groups_Ozero__class_Ozero(T_a)
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_a_2,V_b_2) = c_Groups_Oone__class_Oone(T_a)
<=> V_a_2 = V_b_2 ) ) ) )).
fof(fact_nonzero__power__divide,axiom,(
! [V_n,V_a,V_b,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_b != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)),V_n) = c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) ) )).
fof(fact_power__one__over,axiom,(
! [V_n,V_a,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oone__class_Oone(T_a),V_a)),V_n) ) )).
fof(fact_real__0__le__divide__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_xa_2,V_ya_2))
<=> ( ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
| c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_ya_2) )
& ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_xa_2)
| c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_ya_2,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) ) ) ) )).
fof(fact_divide__eq__eq__number__of1,axiom,(
! [V_a_2,V_w_2,V_b_2,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2)) = V_a_2
<=> ( ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) != c_Groups_Ozero__class_Ozero(T_a)
=> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)) )
& ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = c_Groups_Ozero__class_Ozero(T_a)
=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_divide__eq__eq__number__of,axiom,(
! [V_w_2,V_c_2,V_b_2,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2) = c_Int_Onumber__class_Onumber__of(T_a,V_w_2)
<=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2) )
& ( V_c_2 = c_Groups_Ozero__class_Ozero(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_eq__divide__eq__number__of,axiom,(
! [V_c_2,V_b_2,V_w_2,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2)
<=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2) = V_b_2 )
& ( V_c_2 = c_Groups_Ozero__class_Ozero(T_a)
=> c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_eq__divide__eq__number__of1,axiom,(
! [V_w_2,V_b_2,V_a_2,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( V_a_2 = c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
<=> ( ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)) = V_b_2 )
& ( c_Int_Onumber__class_Onumber__of(T_a,V_w_2) = c_Groups_Ozero__class_Ozero(T_a)
=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_divide__Numeral0,axiom,(
! [V_x,T_a] :
( ( class_Fields_Ofield__inverse__zero(T_a)
& class_Int_Onumber__ring(T_a) )
=> c_Rings_Oinverse__class_Odivide(T_a,V_x,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OPls)) = c_Groups_Ozero__class_Ozero(T_a) ) )).
fof(fact_divide__Numeral1,axiom,(
! [V_x,T_a] :
( ( class_Fields_Ofield(T_a)
& class_Int_Onumber__ring(T_a) )
=> c_Rings_Oinverse__class_Odivide(T_a,V_x,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls))) = V_x ) )).
fof(fact_divide__numeral__1,axiom,(
! [V_a,T_a] :
( ( class_Fields_Ofield(T_a)
& class_Int_Onumber__ring(T_a) )
=> c_Rings_Oinverse__class_Odivide(T_a,V_a,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit1(c_Int_OPls))) = V_a ) )).
fof(fact_power__diff,axiom,(
! [V_m,V_n,V_a,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_a != c_Groups_Ozero__class_Ozero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) )).
fof(fact_real__average__minus__second,axiom,(
! [V_a,V_b] : c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_b,V_a),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b,V_a),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).
fof(fact_real__average__minus__first,axiom,(
! [V_b,V_a] : c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_b),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b,V_a),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).
fof(fact_real__sum__of__halves,axiom,(
! [V_x] : c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_x,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_x,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = V_x )).
fof(fact_Deriv_Oadd__diff__add,axiom,(
! [V_d,V_b,V_c,V_a,T_a] :
( class_Groups_Oab__group__add(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b),c_Groups_Ominus__class_Ominus(T_a,V_c,V_d)) ) )).
fof(fact_lemma__MVT,axiom,(
! [V_b_2,V_a_2,V_f_2] : c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(V_f_2,V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(V_f_2,V_b_2),hAPP(V_f_2,V_a_2)),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b_2,V_a_2))),V_a_2)) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(V_f_2,V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(V_f_2,V_b_2),hAPP(V_f_2,V_a_2)),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b_2,V_a_2))),V_b_2)) )).
fof(fact_DERIV__mult__lemma,axiom,(
! [V_h,V_d,V_c,V_b,V_a,T_a] :
( class_RealVector_Oreal__field(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_d)),V_h) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_d),V_h)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_c),V_h)),V_d)) ) )).
fof(fact_eq__divide__2__times__iff,axiom,(
! [V_za_2,V_ya_2,V_xa_2] :
( V_xa_2 = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_ya_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_za_2))
<=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_xa_2) = c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_ya_2,V_za_2) ) )).
fof(fact_diff__divide__eq__iff,axiom,(
! [V_y,V_x,V_z,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,V_x,c_Rings_Oinverse__class_Odivide(T_a,V_y,V_z)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x),V_y),V_z) ) ) )).
fof(fact_divide__diff__eq__iff,axiom,(
! [V_y,V_x,V_z,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_z),V_y) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_z) ) ) )).
fof(fact_diff__frac__eq,axiom,(
! [V_w,V_x,V_z,V_y,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_y != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Ominus__class_Ominus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Rings_Oinverse__class_Odivide(T_a,V_w,V_z)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Ominus__class_Ominus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) ) ) ) )).
fof(fact_times__divide__times__eq,axiom,(
! [V_w,V_z,V_y,V_x,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)),c_Rings_Oinverse__class_Odivide(T_a,V_z,V_w)) = c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_w)) ) )).
fof(fact_abs__divide,axiom,(
! [V_b,V_a,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)) ) )).
fof(fact_zero__le__divide__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_a_2,V_b_2))
<=> ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) )).
fof(fact_divide__le__0__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a_2,V_b_2),c_Groups_Ozero__class_Ozero(T_a))
<=> ( ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) )
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) ) ) ) ) )).
fof(fact_divide__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) ) ) ) )).
fof(fact_divide__right__mono__neg,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c)) ) ) ) )).
fof(fact_eq__divide__eq,axiom,(
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( V_a_2 = c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2)
<=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2) = V_b_2 )
& ( V_c_2 = c_Groups_Ozero__class_Ozero(T_a)
=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_divide__eq__eq,axiom,(
! [V_a_2,V_c_2,V_b_2,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2) = V_a_2
<=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_a)
=> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2) )
& ( V_c_2 = c_Groups_Ozero__class_Ozero(T_a)
=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) )).
fof(fact_mult__divide__mult__cancel__right,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( V_c != c_Groups_Ozero__class_Ozero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b) ) ) )).
fof(fact_mult__divide__mult__cancel__left,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( V_c != c_Groups_Ozero__class_Ozero(T_a)
=> c_Rings_Oinverse__class_Odivide(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b) ) ) )).
fof(fact_frac__eq__eq,axiom,(
! [V_w_2,V_xa_2,V_za_2,V_ya_2,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_ya_2 != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_za_2 != c_Groups_Ozero__class_Ozero(T_a)
=> ( c_Rings_Oinverse__class_Odivide(T_a,V_xa_2,V_ya_2) = c_Rings_Oinverse__class_Odivide(T_a,V_w_2,V_za_2)
<=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_za_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w_2),V_ya_2) ) ) ) ) )).
fof(fact_nonzero__abs__divide,axiom,(
! [V_a,V_b,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( V_b != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oabs__class_Oabs(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_b)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)) ) ) )).
fof(fact_add__frac__eq,axiom,(
! [V_w,V_x,V_z,V_y,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_y != c_Groups_Ozero__class_Ozero(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Rings_Oinverse__class_Odivide(T_a,V_w,V_z)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) ) ) ) )).
fof(fact_divide__add__eq__iff,axiom,(
! [V_y,V_x,V_z,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_z),V_y) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_z) ) ) )).
fof(fact_add__frac__num,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( V_y != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),V_z) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_y) ) ) )).
fof(fact_add__divide__eq__iff,axiom,(
! [V_y,V_x,V_z,T_a] :
( class_Fields_Ofield(T_a)
=> ( V_z != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_x,c_Rings_Oinverse__class_Odivide(T_a,V_y,V_z)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x),V_y),V_z) ) ) )).
fof(fact_add__num__frac,axiom,(
! [V_x,V_z,V_y,T_a] :
( class_Fields_Ofield__inverse__zero(T_a)
=> ( V_y != c_Groups_Ozero__class_Ozero(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_z,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) = c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_y) ) ) )).
fof(fact_arctan__half,axiom,(
! [V_x] : c_Transcendental_Oarctan(V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Transcendental_Oarctan(c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_x,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))))) )).
fof(fact_lemma__sqrt__hcomplex__capprox,axiom,(
! [V_y,V_x,V_u] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_u)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_u,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_y,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_u,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),V_u) ) ) ) ) ) )).
fof(fact_real__sqrt__sum__squares__less,axiom,(
! [V_y,V_u,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_u,c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_y),c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_u,c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),V_u) ) ) )).
fof(fact_mult__strict__left__mono__neg,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_mult__strict__right__mono__neg,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) ) ) )).
fof(fact_comm__mult__strict__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__comm__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_mult__strict__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) ) ) ) )).
fof(fact_mult__strict__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) ) ) )).
fof(fact_mult__neg__neg,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) ) ) ) )).
fof(fact_mult__neg__pos,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__less__cancel__left__neg,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) ) ) ) )).
fof(fact_zero__less__mult__pos2,axiom,(
! [V_a,V_b,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) ) ) ) )).
fof(fact_zero__less__mult__pos,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) ) ) ) )).
fof(fact_mult__pos__neg2,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__pos__neg,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_mult__pos__pos,axiom,(
! [V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) ) ) ) )).
fof(fact_mult__less__cancel__left__pos,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) ) )).
fof(fact_mult__less__cancel__left__disj,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
& c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) )
| ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) ) ) ) ) )).
fof(fact_mult__less__cancel__right__disj,axiom,(
! [V_b_2,V_c_2,V_a_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_c_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
& c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) )
| ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) ) ) ) ) )).
fof(fact_not__square__less__zero,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__ring(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_add__neg__neg,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_add__pos__pos,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) )).
fof(fact_double__add__less__zero__iff__single__add__less__zero,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_zero__less__double__add__iff__zero__less__single__add,axiom,(
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))
<=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) )).
fof(fact_even__less__0__iff,axiom,(
! [V_a_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_pos__add__strict,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) )).
fof(fact_arctan__zero__zero,axiom,(
c_Transcendental_Oarctan(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) )).
fof(fact_arctan__monotone_H,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Transcendental_Oarctan(V_x),c_Transcendental_Oarctan(V_y)) ) )).
fof(fact_arctan__monotone,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Transcendental_Oarctan(V_x),c_Transcendental_Oarctan(V_y)) ) )).
fof(fact_real__sqrt__less__mono,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_x),c_NthRoot_Osqrt(V_y)) ) )).
fof(fact_real__sqrt__less__iff,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,V_ya_2) ) )).
fof(fact_less__special_I3_J,axiom,(
! [V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_xa_2,c_Int_OPls) ) ) )).
fof(fact_less__special_I1_J,axiom,(
! [V_ya_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,V_ya_2) ) ) )).
fof(fact_power__strict__decreasing,axiom,(
! [V_a,V_N,V_n,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_N)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) ) )).
fof(fact_one__less__power,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) )).
fof(fact_order__less__irrefl,axiom,(
! [V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_x) ) )).
fof(fact_linorder__neq__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( V_xa_2 != V_ya_2
<=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
| c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) ) ) ) )).
fof(fact_not__less__iff__gr__or__eq,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)
| V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_linorder__less__linear,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
| V_x = V_y
| c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) )).
fof(fact_linorder__antisym__conv3,axiom,(
! [V_xa_2,V_ya_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_linorder__neqE,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( V_x != V_y
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ) )).
fof(fact_less__imp__neq,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_x != V_y ) ) )).
fof(fact_order__less__not__sym,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) )).
fof(fact_order__less__imp__not__less,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) )).
fof(fact_order__less__imp__not__eq,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_x != V_y ) ) )).
fof(fact_order__less__imp__not__eq2,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_y != V_x ) ) )).
fof(fact_order__less__asym_H,axiom,(
! [V_b,V_a,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) )).
fof(fact_xt1_I9_J,axiom,(
! [V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) )).
fof(fact_ord__eq__less__trans,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_c) ) ) ) )).
fof(fact_xt1_I1_J,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_c,V_a) ) ) ) )).
fof(fact_ord__less__eq__trans,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_c) ) ) ) )).
fof(fact_xt1_I2_J,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless(T_a,V_c,V_a) ) ) ) )).
fof(fact_order__less__trans,axiom,(
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) )).
fof(fact_xt1_I10_J,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) )).
fof(fact_order__less__asym,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) )).
fof(fact_linorder__cases,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( V_x != V_y
=> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ) )).
fof(fact_less__eq__real__def,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,V_ya_2)
| V_xa_2 = V_ya_2 ) ) )).
fof(fact_real__less__def,axiom,(
! [V_ya_2,V_xa_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2)
& V_xa_2 != V_ya_2 ) ) )).
fof(fact_diff__eq__diff__less,axiom,(
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)
<=> c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2) ) ) ) )).
fof(fact_add__less__cancel__right,axiom,(
! [V_b_2,V_c_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) )).
fof(fact_add__less__cancel__left,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) )).
fof(fact_add__strict__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ) )).
fof(fact_add__strict__left__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) ) ) )).
fof(fact_add__strict__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) )).
fof(fact_add__less__imp__less__right,axiom,(
! [V_b,V_c,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) )).
fof(fact_add__less__imp__less__left,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) )).
fof(fact_xt1_I8_J,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) )).
fof(fact_order__le__less__trans,axiom,(
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) )).
fof(fact_xt1_I7_J,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) )).
fof(fact_order__less__le__trans,axiom,(
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) )).
fof(fact_xt1_I11_J,axiom,(
! [V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( V_a != V_b
=> c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) ) )).
fof(fact_order__le__neq__trans,axiom,(
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( V_a != V_b
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_order__le__imp__less__or__eq,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
| V_x = V_y ) ) ) )).
fof(fact_linorder__antisym__conv2,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_order__less__imp__le,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) )).
fof(fact_leD,axiom,(
! [V_x,V_y,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y) ) ) )).
fof(fact_xt1_I12_J,axiom,(
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a != V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) ) )).
fof(fact_order__neq__le__trans,axiom,(
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a != V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_linorder__antisym__conv1,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
<=> V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_not__leE,axiom,(
! [V_x,V_y,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_y) ) ) )).
fof(fact_leI,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) )).
fof(fact_order__le__less,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
| V_xa_2 = V_ya_2 ) ) ) )).
fof(fact_less__le__not__le,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
& ~ c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) ) )).
fof(fact_order__less__le,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
& V_xa_2 != V_ya_2 ) ) ) )).
fof(fact_linorder__le__less__linear,axiom,(
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
| c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) )).
fof(fact_linorder__not__le,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
<=> c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) ) ) )).
fof(fact_linorder__not__less,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) )).
fof(fact_power__strict__increasing,axiom,(
! [V_a,V_N,V_n,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_N)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)) ) ) ) )).
fof(fact_power__less__imp__less__exp,axiom,(
! [V_n,V_m,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ) ) )).
fof(fact_power__strict__increasing__iff,axiom,(
! [V_ya_2,V_xa_2,V_b_2,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_b_2)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_xa_2),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_xa_2,V_ya_2) ) ) ) )).
fof(fact_linorder__neqE__linordered__idom,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( V_x != V_y
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ) )).
fof(fact_less__number__of,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_xa_2,V_ya_2) ) ) )).
fof(fact_less__special_I4_J,axiom,(
! [V_xa_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_xa_2),c_Groups_Oone__class_Oone(T_a))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_xa_2,c_Int_OBit1(c_Int_OPls)) ) ) )).
fof(fact_less__special_I2_J,axiom,(
! [V_ya_2,T_a] :
( ( class_Int_Onumber__ring(T_a)
& class_Rings_Olinordered__idom(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(c_Int_OPls),V_ya_2) ) ) )).
fof(fact_power__strict__mono,axiom,(
! [V_n,V_b,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) ) ) ) )).
fof(fact_add__le__less__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) )).
fof(fact_add__less__le__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) )).
fof(fact_le__number__of__eq__not__less,axiom,(
! [V_w_2,V_v_2,T_a] :
( ( class_Int_Onumber(T_a)
& class_Orderings_Olinorder(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_v_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
<=> ~ c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Int_Onumber__class_Onumber__of(T_a,V_v_2)) ) ) )).
fof(fact_zero__less__one,axiom,(
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)) ) )).
fof(fact_not__one__less__zero,axiom,(
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_less__iff__diff__less__0,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)
<=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2),c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_less__1__mult,axiom,(
! [V_n,V_m,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_m)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_n)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),V_n)) ) ) ) )).
fof(fact_zero__less__power,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) )).
fof(fact_less__add__one,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))) ) )).
fof(fact_divide__strict__right__mono__neg,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) ) ) ) )).
fof(fact_divide__strict__right__mono,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a,V_c),c_Rings_Oinverse__class_Odivide(T_a,V_b,V_c)) ) ) ) )).
fof(fact_divide__neg__neg,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_divide__neg__pos,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_divide__pos__neg,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_divide__pos__pos,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_divide__less__0__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_a_2,V_b_2),c_Groups_Ozero__class_Ozero(T_a))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) )
| ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) ) ) ) ) )).
fof(fact_zero__less__divide__iff,axiom,(
! [V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_a_2,V_b_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)
& c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) )
| ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))
& c_Orderings_Oord__class_Oless(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) )).
fof(fact_power__inject__exp,axiom,(
! [V_n_2,V_m_2,V_a_2,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2)
=> ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_m_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2)
<=> V_m_2 = V_n_2 ) ) ) )).
fof(fact_abs__not__less__zero,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_zero__less__abs__iff,axiom,(
! [V_a_2,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a_2))
<=> V_a_2 != c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_abs__of__pos,axiom,(
! [V_a,T_a] :
( class_Groups_Oordered__ab__group__add__abs(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> c_Groups_Oabs__class_Oabs(T_a,V_a) = V_a ) ) )).
fof(fact_abs__mult__less,axiom,(
! [V_d,V_b,V_c,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_c)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_b),V_d)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_d)) ) ) ) )).
fof(fact_real__mult__less__mono2,axiom,(
! [V_y,V_x,V_z] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_z)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_y)) ) ) )).
fof(fact_real__mult__order,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y)) ) ) )).
fof(fact_real__mult__less__iff1,axiom,(
! [V_ya_2,V_xa_2,V_za_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_xa_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_ya_2),V_za_2))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,V_ya_2) ) ) )).
fof(fact_not__real__square__gt__zero,axiom,(
! [V_xa_2] :
( ~ c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_xa_2),V_xa_2))
<=> V_xa_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_norm__not__less__zero,axiom,(
! [V_x,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ~ c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) ) )).
fof(fact_real__sqrt__gt__0__iff,axiom,(
! [V_ya_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_ya_2) ) )).
fof(fact_real__sqrt__lt__0__iff,axiom,(
! [V_xa_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) ) )).
fof(fact_real__sqrt__not__eq__zero,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> c_NthRoot_Osqrt(V_x) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) ) )).
fof(fact_real__sqrt__gt__zero,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)) ) )).
fof(fact_real__sqrt__lt__1__iff,axiom,(
! [V_xa_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_NthRoot_Osqrt(V_xa_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_xa_2,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) ) )).
fof(fact_real__sqrt__gt__1__iff,axiom,(
! [V_ya_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_ya_2))
<=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_ya_2) ) )).
fof(fact_mult__le__cancel__left__pos,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) ) )).
fof(fact_mult__le__cancel__left__neg,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2) ) ) ) )).
fof(fact_mult__strict__mono,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_mult__strict__mono_H,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_mult__less__le__imp__less,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_mult__le__less__imp__less,axiom,(
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d)) ) ) ) ) ) )).
fof(fact_mult__right__less__imp__less,axiom,(
! [V_b,V_c,V_a,T_a] :
( class_Rings_Olinordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_mult__less__imp__less__right,axiom,(
! [V_b,V_c,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_mult__left__less__imp__less,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Rings_Olinordered__semiring(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_mult__less__imp__less__left,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_mult__right__le__imp__le,axiom,(
! [V_b,V_c,V_a,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) ) )).
fof(fact_mult__left__le__imp__le,axiom,(
! [V_b,V_a,V_c,T_a] :
( class_Rings_Olinordered__semiring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) ) )).
fof(fact_add__nonpos__neg,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_add__neg__nonpos,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_add__strict__increasing2,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) )).
fof(fact_add__strict__increasing,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) )).
fof(fact_add__nonneg__pos,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) )).
fof(fact_add__pos__nonneg,axiom,(
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) )).
fof(fact_not__sum__squares__lt__zero,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__ring(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y)),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_sum__squares__gt__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__ring__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa_2),V_xa_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ya_2),V_ya_2)))
<=> ( V_xa_2 != c_Groups_Ozero__class_Ozero(T_a)
| V_ya_2 != c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_power__less__imp__less__base,axiom,(
! [V_b,V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) )).
fof(fact_zero__less__two,axiom,(
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))) ) )).
fof(fact_divide__nonneg__pos,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_divide__nonneg__neg,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_frac__less2,axiom,(
! [V_z,V_w,V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_w)
=> ( c_Orderings_Oord__class_Oless(T_a,V_w,V_z)
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_z),c_Rings_Oinverse__class_Odivide(T_a,V_y,V_w)) ) ) ) ) ) )).
fof(fact_frac__less,axiom,(
! [V_z,V_w,V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_w)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_w,V_z)
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_z),c_Rings_Oinverse__class_Odivide(T_a,V_y,V_w)) ) ) ) ) ) )).
fof(fact_frac__le,axiom,(
! [V_z,V_w,V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_w)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_w,V_z)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_z),c_Rings_Oinverse__class_Odivide(T_a,V_y,V_w)) ) ) ) ) ) )).
fof(fact_divide__nonpos__pos,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) )).
fof(fact_divide__nonpos__neg,axiom,(
! [V_y,V_x,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_less__add__iff2,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2)) ) ) )).
fof(fact_less__add__iff1,axiom,(
! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_a] :
( class_Rings_Oordered__ring(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2),V_d_2))
<=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2),V_d_2) ) ) )).
fof(fact_less__divide__eq,axiom,(
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_divide__less__eq,axiom,(
! [V_a_2,V_c_2,V_b_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ) ) ) ) )).
fof(fact_pos__less__divide__eq,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) ) ) ) )).
fof(fact_pos__divide__less__eq,axiom,(
! [V_a_2,V_b_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) ) ) ) )).
fof(fact_mult__imp__div__pos__less,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),V_z) ) ) ) )).
fof(fact_mult__imp__less__div__pos,axiom,(
! [V_x,V_z,V_y,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y),V_x)
=> c_Orderings_Oord__class_Oless(T_a,V_z,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_neg__less__divide__eq,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) ) ) ) )).
fof(fact_neg__divide__less__eq,axiom,(
! [V_a_2,V_b_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) ) ) ) )).
fof(fact_divide__strict__left__mono,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_c,V_a),c_Rings_Oinverse__class_Odivide(T_a,V_c,V_b)) ) ) ) ) )).
fof(fact_divide__strict__left__mono__neg,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_c,V_a),c_Rings_Oinverse__class_Odivide(T_a,V_c,V_b)) ) ) ) ) )).
fof(fact_power__gt1__lemma,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) )).
fof(fact_power__less__power__Suc,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) )).
fof(fact_less__half__sum,axiom,(
! [V_b,V_a,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_a,c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a)))) ) ) )).
fof(fact_gt__half__sum,axiom,(
! [V_b,V_a,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))),V_b) ) ) )).
fof(fact_power__gt1,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n))) ) ) )).
fof(fact_abs__diff__less__iff,axiom,(
! [V_r_2,V_a_2,V_xa_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_xa_2,V_a_2)),V_r_2)
<=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_r_2),V_xa_2)
& c_Orderings_Oord__class_Oless(T_a,V_xa_2,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_r_2)) ) ) ) )).
fof(fact_abs__div__pos,axiom,(
! [V_x,V_y,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oabs__class_Oabs(T_a,V_x),V_y) = c_Groups_Oabs__class_Oabs(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) )).
fof(fact_zero__less__norm__iff,axiom,(
! [V_xa_2,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_xa_2))
<=> V_xa_2 != c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_real__mult__le__cancel__iff1,axiom,(
! [V_ya_2,V_xa_2,V_za_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_xa_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_ya_2),V_za_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2) ) ) )).
fof(fact_real__mult__le__cancel__iff2,axiom,(
! [V_ya_2,V_xa_2,V_za_2] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_za_2),V_xa_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_za_2),V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_xa_2,V_ya_2) ) ) )).
fof(fact_norm__mult__less,axiom,(
! [V_s,V_y,V_r,V_x,T_a] :
( class_RealVector_Oreal__normed__algebra(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),V_r)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_y),V_s)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_s)) ) ) ) )).
fof(fact_norm__add__less,axiom,(
! [V_s,V_y,V_r,V_x,T_a] :
( class_RealVector_Oreal__normed__vector(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),V_r)
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_y),V_s)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_r,V_s)) ) ) ) )).
fof(fact_abs__add__one__not__less__self,axiom,(
! [V_x] : ~ c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x) )).
fof(fact_arctan__add,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))
=> ( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_y),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))
=> c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Transcendental_Oarctan(V_x),c_Transcendental_Oarctan(V_y)) = c_Transcendental_Oarctan(c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y)))) ) ) )).
fof(fact_divide__left__mono__neg,axiom,(
! [V_c,V_b,V_a,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_c,V_a),c_Rings_Oinverse__class_Odivide(T_a,V_c,V_b)) ) ) ) ) )).
fof(fact_divide__left__mono,axiom,(
! [V_c,V_a,V_b,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_c,V_a),c_Rings_Oinverse__class_Odivide(T_a,V_c,V_b)) ) ) ) ) )).
fof(fact_neg__divide__le__eq,axiom,(
! [V_a_2,V_b_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) ) ) ) )).
fof(fact_neg__le__divide__eq,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) ) ) ) )).
fof(fact_mult__imp__le__div__pos,axiom,(
! [V_x,V_z,V_y,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y),V_x)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_z,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y)) ) ) ) )).
fof(fact_mult__imp__div__pos__le,axiom,(
! [V_z,V_x,V_y,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_x,V_y),V_z) ) ) ) )).
fof(fact_pos__divide__le__eq,axiom,(
! [V_a_2,V_b_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) ) ) ) )).
fof(fact_pos__le__divide__eq,axiom,(
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Fields_Olinordered__field(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) ) ) ) )).
fof(fact_divide__le__eq,axiom,(
! [V_a_2,V_c_2,V_b_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),V_a_2)
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ) ) ) ) )).
fof(fact_le__divide__eq,axiom,(
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Fields_Olinordered__field__inverse__zero(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_power__Suc__less,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) )).
fof(fact_divide__less__eq__number__of1,axiom,(
! [V_a_2,V_w_2,V_b_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_a_2)
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ) ) ) ) )).
fof(fact_divide__less__eq__number__of,axiom,(
! [V_w_2,V_c_2,V_b_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)) ) ) ) ) ) ) )).
fof(fact_less__divide__eq__number__of,axiom,(
! [V_c_2,V_b_2,V_w_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_less__divide__eq__number__of1,axiom,(
! [V_w_2,V_b_2,V_a_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2)))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_power__Suc__less__one,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oone__class_Oone(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)),c_Groups_Oone__class_Oone(T_a)) ) ) ) )).
fof(fact_power__increasing__iff,axiom,(
! [V_ya_2,V_xa_2,V_b_2,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_b_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_xa_2),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_ya_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_xa_2,V_ya_2) ) ) ) )).
fof(fact_power__le__imp__le__exp,axiom,(
! [V_n,V_m,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ) ) )).
fof(fact_abs__add__one__gt__zero,axiom,(
! [V_x] : c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x))) )).
fof(fact_convex__bound__lt,axiom,(
! [V_v,V_u,V_y,V_a,V_x,T_a] :
( class_Rings_Olinordered__semiring__1__strict(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_u)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_v)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_u,V_v) = c_Groups_Oone__class_Oone(T_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y)),V_a) ) ) ) ) ) ) )).
fof(fact_divide__le__eq__number__of1,axiom,(
! [V_a_2,V_w_2,V_b_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_a_2)
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ) ) ) ) )).
fof(fact_divide__le__eq__number__of,axiom,(
! [V_w_2,V_c_2,V_b_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)) ) ) ) ) ) ) )).
fof(fact_le__divide__eq__number__of,axiom,(
! [V_c_2,V_b_2,V_w_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Rings_Oinverse__class_Odivide(T_a,V_b_2,V_c_2))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2)
=> ( ( c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_c_2)) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_le__divide__eq__number__of1,axiom,(
! [V_w_2,V_b_2,V_a_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber(T_a) )
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Rings_Oinverse__class_Odivide(T_a,V_b_2,c_Int_Onumber__class_Onumber__of(T_a,V_w_2)))
<=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2)),V_b_2) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))
=> ( ( c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(T_a,V_w_2))) )
& ( ~ c_Orderings_Oord__class_Oless(T_a,c_Int_Onumber__class_Onumber__of(T_a,V_w_2),c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ) ) ) )).
fof(fact_half__gt__zero__iff,axiom,(
! [V_r_2,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber__ring(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_r_2,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
<=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_r_2) ) ) )).
fof(fact_half__gt__zero,axiom,(
! [V_r,T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber__ring(T_a) )
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_r)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,V_r,c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) ) )).
fof(fact_power2__less__0,axiom,(
! [V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_zero__less__power2,axiom,(
! [V_a_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
<=> V_a_2 != c_Groups_Ozero__class_Ozero(T_a) ) ) )).
fof(fact_real__sqrt__two__gt__zero,axiom,(
c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).
fof(fact_real__less__half__sum,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).
fof(fact_real__gt__half__sum,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_y) ) )).
fof(fact_real__sqrt__two__gt__one,axiom,(
c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).
fof(fact_power2__less__imp__less,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_y) ) ) ) )).
fof(fact_half,axiom,(
! [T_a] :
( ( class_Fields_Olinordered__field__inverse__zero(T_a)
& class_Int_Onumber__ring(T_a) )
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Rings_Oinverse__class_Odivide(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Onumber__class_Onumber__of(T_a,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).
fof(fact_not__sum__power2__lt__zero,axiom,(
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(T_a)) ) )).
fof(fact_sum__power2__gt__zero__iff,axiom,(
! [V_ya_2,V_xa_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_xa_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_ya_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
<=> ( V_xa_2 != c_Groups_Ozero__class_Ozero(T_a)
| V_ya_2 != c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).
fof(fact_real__sqrt__pow2__gt__zero,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_NthRoot_Osqrt(V_x)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).
fof(fact_lemma__real__divide__sqrt__less,axiom,(
! [V_u] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_u)
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,V_u,c_NthRoot_Osqrt(c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),V_u) ) )).
fof(fact_less__one__imp__sqr__less__one,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))
=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) ) )).
fof(fact_odd__power__less__zero,axiom,(
! [V_n,V_a,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))),c_Groups_Ozero__class_Ozero(T_a)) ) ) )).
fof(fact_machin,axiom,(
c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Transcendental_Opi,c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Transcendental_Oarctan(c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit1(c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))),c_Transcendental_Oarctan(c_Rings_Oinverse__class_Odivide(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit1(c_Int_OBit1(c_Int_OBit1(c_Int_OBit1(c_Int_OBit0(c_Int_OBit1(c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))))))))))) )).
fof(fact_complex_Orecs,axiom,(
! [V_real2_2,V_real1_2,V_f1_2,T_a] : c_Complex_Ocomplex_Ocomplex__rec(T_a,V_f1_2,c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2)) = hAPP(hAPP(V_f1_2,V_real1_2),V_real2_2) )).
fof(fact_natceiling__add__one,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x)
=> c_RComplete_Onatceiling(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_RComplete_Onatceiling(V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) ) )).
fof(fact_less__zeroE,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_Suc__mono,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n)) ) )).
fof(fact_lessI,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Nat_OSuc(V_n)) )).
fof(fact_zero__less__Suc,axiom,(
! [V_n] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(V_n)) )).
fof(fact_add__lessD1,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_k) ) )).
fof(fact_less__add__eq__less,axiom,(
! [V_n,V_m,V_l,V_k] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l)
=> ( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_l) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ) )).
fof(fact_add__less__mono,axiom,(
! [V_l,V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l)) ) ) )).
fof(fact_add__less__mono1,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)) ) )).
fof(fact_trans__less__add2,axiom,(
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)) ) )).
fof(fact_trans__less__add1,axiom,(
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)) ) )).
fof(fact_nat__add__left__cancel__less,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_not__add__less2,axiom,(
! [V_i,V_j] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_i),V_i) )).
fof(fact_not__add__less1,axiom,(
! [V_j,V_i] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_i) )).
fof(fact_pi__not__less__zero,axiom,(
~ c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Transcendental_Opi,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) )).
fof(fact_not__less0,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_pi__gt__zero,axiom,(
c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Opi) )).
fof(fact_neq0__conv,axiom,(
! [V_n_2] :
( V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) ) )).
fof(fact_less__nat__zero__code,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).
fof(fact_gr__implies__not0,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).
fof(fact_less__imp__diff__less,axiom,(
! [V_n,V_k,V_j] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_n),V_k) ) )).
fof(fact_diff__less__mono2,axiom,(
! [V_l,V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_l)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_n),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_m)) ) ) )).
fof(fact_gr0I,axiom,(
! [V_n] :
( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) ) )).
fof(fact_nat__less__le,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
& V_m_2 != V_n_2 ) ) )).
fof(fact_le__eq__less__or__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
| V_m_2 = V_n_2 ) ) )).
fof(fact_less__imp__le__nat,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_le__neq__implies__less,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ( V_m != V_n
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ) )).
fof(fact_less__or__eq__imp__le,axiom,(
! [V_n,V_m] :
( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
| V_m = V_n )
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_less__not__refl,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n) )).
fof(fact_nat__neq__iff,axiom,(
! [V_n_2,V_m_2] :
( V_m_2 != V_n_2
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) ) ) )).
fof(fact_linorder__neqE__nat,axiom,(
! [V_y,V_x] :
( V_x != V_y
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_y,V_x) ) ) )).
fof(fact_less__irrefl__nat,axiom,(
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n) )).
fof(fact_less__not__refl2,axiom,(
! [V_m,V_n] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m)
=> V_m != V_n ) )).
fof(fact_less__not__refl3,axiom,(
! [V_t,V_s] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s,V_t)
=> V_s != V_t ) )).
fof(fact_nat__less__cases,axiom,(
! [V_P_2,V_n_2,V_m_2] :
( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> ( ( V_m_2 = V_n_2
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> ( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) ) ) ) )).
fof(fact_Suc__less__SucD,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n))
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_Suc__lessD,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_less__SucE,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n))
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> V_m = V_n ) ) )).
fof(fact_less__trans__Suc,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_i),V_k) ) ) )).
fof(fact_Suc__lessI,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> ( c_Nat_OSuc(V_m) != V_n
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) ) ) )).
fof(fact_less__SucI,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) )).
fof(fact_less__antisym,axiom,(
! [V_m,V_n] :
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Nat_OSuc(V_m))
=> V_m = V_n ) ) )).
fof(fact_not__less__less__Suc__eq,axiom,(
! [V_m_2,V_n_2] :
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2))
<=> V_n_2 = V_m_2 ) ) )).
fof(fact_Suc__less__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m_2),c_Nat_OSuc(V_n_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_less__Suc__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
| V_m_2 = V_n_2 ) ) )).
fof(fact_not__less__eq,axiom,(
! [V_n_2,V_m_2] :
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2)) ) )).
fof(fact_rel__simps_I7_J,axiom,(
~ c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OMin,c_Int_OMin) )).
fof(fact_zless__linear,axiom,(
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,V_x,V_y)
| V_x = V_y
| c_Orderings_Oord__class_Oless(tc_Int_Oint,V_y,V_x) ) )).
fof(fact_less__number__of__int__code,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_k_2),c_Int_Onumber__class_Onumber__of(tc_Int_Oint,V_l_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_zadd__strict__right__mono,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_j,V_k)) ) )).
fof(fact_zless__le,axiom,(
! [V_w_2,V_za_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,V_za_2,V_w_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_za_2,V_w_2)
& V_za_2 != V_w_2 ) ) )).
fof(fact_less__int__code_I13_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit0(V_k1_2),c_Int_OBit0(V_k2_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I14_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OBit0(V_l_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_rel__simps_I2_J,axiom,(
~ c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,c_Int_OPls) )).
fof(fact_less__int__code_I16_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_k1_2),c_Int_OBit1(V_k2_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I17_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_k_2),c_Int_OBit1(V_l_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_less__nat__number__of,axiom,(
! [V_v_H_2,V_v_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_2),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_H_2))
<=> ( ( c_Orderings_Oord__class_Oless(tc_Int_Oint,V_v_2,V_v_H_2)
=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,V_v_H_2) )
& c_Orderings_Oord__class_Oless(tc_Int_Oint,V_v_2,V_v_H_2) ) ) )).
fof(fact_less__fun__def,axiom,(
! [V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b),V_f_2,V_g_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
& ~ c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2,V_f_2) ) ) ) )).
fof(fact_less__0__number__of,axiom,(
! [V_v_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,V_v_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,V_v_2) ) )).
fof(fact_pi__ge__zero,axiom,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Opi) )).
fof(fact_pi__neq__zero,axiom,(
c_Transcendental_Opi != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) )).
fof(fact_bin__less__0__simps_I4_J,axiom,(
! [V_w_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_w_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) ) )).
fof(fact_rel__simps_I12_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_k_2),c_Int_OPls)
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,c_Int_OPls) ) )).
fof(fact_bin__less__0__simps_I1_J,axiom,(
~ c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )).
fof(fact_less__int__code_I15_J,axiom,(
! [V_k2_2,V_k1_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_k1_2),c_Int_OBit0(V_k2_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k1_2,V_k2_2) ) )).
fof(fact_rel__simps_I16_J,axiom,(
! [V_l_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit1(V_k_2),c_Int_OBit0(V_l_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,V_l_2) ) )).
fof(fact_bin__less__0__simps_I3_J,axiom,(
! [V_w_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit0(V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_w_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) ) )).
fof(fact_less__Suc__eq__0__disj,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2))
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| ? [B_j] :
( V_m_2 = c_Nat_OSuc(B_j)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_j,V_n_2) ) ) ) )).
fof(fact_less__Suc0,axiom,(
! [V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))
<=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).
fof(fact_gr0__conv__Suc,axiom,(
! [V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)
<=> ? [B_m] : V_n_2 = c_Nat_OSuc(B_m) ) )).
fof(fact_nat__lt__two__imp__zero__or__one,axiom,(
! [V_x] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))
=> ( V_x = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| V_x = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).
fof(fact_rel__simps_I10_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OBit0(V_k_2),c_Int_OPls)
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,c_Int_OPls) ) )).
fof(fact_rel__simps_I4_J,axiom,(
! [V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,c_Int_OBit0(V_k_2))
<=> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Int_OPls,V_k_2) ) )).
fof(fact_add__gr__0,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) ) ) )).
fof(fact_less__add__Suc1,axiom,(
! [V_m,V_i] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_m))) )).
fof(fact_less__add__Suc2,axiom,(
! [V_m,V_i] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_i))) )).
fof(fact_less__iff__Suc__add,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
<=> ? [B_k] : V_n_2 = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k)) ) )).
fof(fact_Suc__le__lessD,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) )).
fof(fact_le__less__Suc__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2))
<=> V_n_2 = V_m_2 ) ) )).
fof(fact_Suc__leI,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) ) )).
fof(fact_le__imp__less__Suc,axiom,(
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) ) )).
fof(fact_Suc__le__eq,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m_2),V_n_2)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_less__Suc__eq__le,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) )).
fof(fact_less__eq__Suc__le,axiom,(
! [V_m_2,V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),V_m_2) ) )).
fof(fact_int__0__less__1,axiom,(
c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) )).
fof(fact_nat__0__less__mult__iff,axiom,(
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) ) ) )).
fof(fact_mult__less__cancel1,axiom,(
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ) )).
fof(fact_mult__less__cancel2,axiom,(
! [V_n_2,V_k_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ) )).
fof(fact_mult__less__mono1,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k)) ) ) )).
fof(fact_mult__less__mono2,axiom,(
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j)) ) ) )).
%----Arity declarations (204)
fof(arity_fun__Orderings_Opreorder,axiom,(
! [T_2,T_1] :
( class_Orderings_Opreorder(T_1)
=> class_Orderings_Opreorder(tc_fun(T_2,T_1)) ) )).
fof(arity_fun__Orderings_Oorder,axiom,(
! [T_2,T_1] :
( class_Orderings_Oorder(T_1)
=> class_Orderings_Oorder(tc_fun(T_2,T_1)) ) )).
fof(arity_fun__Orderings_Oord,axiom,(
! [T_2,T_1] :
( class_Orderings_Oord(T_1)
=> class_Orderings_Oord(tc_fun(T_2,T_1)) ) )).
fof(arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,(
class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add,axiom,(
class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le,axiom,(
class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__comm__semiring__strict,axiom,(
class_Rings_Olinordered__comm__semiring__strict(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__semiring__1__strict,axiom,(
class_Rings_Olinordered__semiring__1__strict(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__semiring__strict,axiom,(
class_Rings_Olinordered__semiring__strict(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__ab__semigroup__add,axiom,(
class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__ab__group__add__abs,axiom,(
class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__comm__monoid__add,axiom,(
class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Olinordered__ab__group__add,axiom,(
class_Groups_Olinordered__ab__group__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add,axiom,(
class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oring__1__no__zero__divisors,axiom,(
class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oordered__cancel__semiring,axiom,(
class_Rings_Oordered__cancel__semiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__ring__strict,axiom,(
class_Rings_Olinordered__ring__strict(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oring__no__zero__divisors,axiom,(
class_Rings_Oring__no__zero__divisors(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oordered__comm__semiring,axiom,(
class_Rings_Oordered__comm__semiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__semiring__1,axiom,(
class_Rings_Olinordered__semiring__1(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oordered__ab__group__add,axiom,(
class_Groups_Oordered__ab__group__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ocancel__semigroup__add,axiom,(
class_Groups_Ocancel__semigroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__semiring,axiom,(
class_Rings_Olinordered__semiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__semidom,axiom,(
class_Rings_Olinordered__semidom(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oab__semigroup__mult,axiom,(
class_Groups_Oab__semigroup__mult(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ocomm__monoid__mult,axiom,(
class_Groups_Ocomm__monoid__mult(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oab__semigroup__add,axiom,(
class_Groups_Oab__semigroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oordered__semiring,axiom,(
class_Rings_Oordered__semiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oordered__ring__abs,axiom,(
class_Rings_Oordered__ring__abs(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Ono__zero__divisors,axiom,(
class_Rings_Ono__zero__divisors(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ocomm__monoid__add,axiom,(
class_Groups_Ocomm__monoid__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__ring,axiom,(
class_Rings_Olinordered__ring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Olinordered__idom,axiom,(
class_Rings_Olinordered__idom(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Ocomm__semiring__1,axiom,(
class_Rings_Ocomm__semiring__1(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Ocomm__semiring,axiom,(
class_Rings_Ocomm__semiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oab__group__add,axiom,(
class_Groups_Oab__group__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Ozero__neq__one,axiom,(
class_Rings_Ozero__neq__one(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oordered__ring,axiom,(
class_Rings_Oordered__ring(tc_Int_Oint) )).
fof(arity_Int__Oint__Orderings_Opreorder,axiom,(
class_Orderings_Opreorder(tc_Int_Oint) )).
fof(arity_Int__Oint__Orderings_Olinorder,axiom,(
class_Orderings_Olinorder(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Omonoid__mult,axiom,(
class_Groups_Omonoid__mult(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Ocomm__ring__1,axiom,(
class_Rings_Ocomm__ring__1(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Omonoid__add,axiom,(
class_Groups_Omonoid__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Osemiring__1,axiom,(
class_Rings_Osemiring__1(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Osemiring__0,axiom,(
class_Rings_Osemiring__0(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ogroup__add,axiom,(
class_Groups_Ogroup__add(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Omult__zero,axiom,(
class_Rings_Omult__zero(tc_Int_Oint) )).
fof(arity_Int__Oint__Orderings_Oorder,axiom,(
class_Orderings_Oorder(tc_Int_Oint) )).
fof(arity_Int__Oint__Int_Oring__char__0,axiom,(
class_Int_Oring__char__0(tc_Int_Oint) )).
fof(arity_Int__Oint__Int_Onumber__ring,axiom,(
class_Int_Onumber__ring(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Osemiring,axiom,(
class_Rings_Osemiring(tc_Int_Oint) )).
fof(arity_Int__Oint__Orderings_Oord,axiom,(
class_Orderings_Oord(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oring__1,axiom,(
class_Rings_Oring__1(tc_Int_Oint) )).
fof(arity_Int__Oint__Power_Opower,axiom,(
class_Power_Opower(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Ozero,axiom,(
class_Groups_Ozero(tc_Int_Oint) )).
fof(arity_Int__Oint__Rings_Oring,axiom,(
class_Rings_Oring(tc_Int_Oint) )).
fof(arity_Int__Oint__Int_Onumber,axiom,(
class_Int_Onumber(tc_Int_Oint) )).
fof(arity_Int__Oint__Groups_Oone,axiom,(
class_Groups_Oone(tc_Int_Oint) )).
fof(arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,(
class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add,axiom,(
class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le,axiom,(
class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict,axiom,(
class_Rings_Olinordered__comm__semiring__strict(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Olinordered__semiring__strict,axiom,(
class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add,axiom,(
class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oordered__comm__monoid__add,axiom,(
class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add,axiom,(
class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Oordered__cancel__semiring,axiom,(
class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Oordered__comm__semiring,axiom,(
class_Rings_Oordered__comm__semiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Ocancel__semigroup__add,axiom,(
class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Olinordered__semiring,axiom,(
class_Rings_Olinordered__semiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Olinordered__semidom,axiom,(
class_Rings_Olinordered__semidom(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oab__semigroup__mult,axiom,(
class_Groups_Oab__semigroup__mult(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Ocomm__monoid__mult,axiom,(
class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oab__semigroup__add,axiom,(
class_Groups_Oab__semigroup__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Oordered__semiring,axiom,(
class_Rings_Oordered__semiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Ono__zero__divisors,axiom,(
class_Rings_Ono__zero__divisors(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Ocomm__monoid__add,axiom,(
class_Groups_Ocomm__monoid__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Ocomm__semiring__1,axiom,(
class_Rings_Ocomm__semiring__1(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Ocomm__semiring,axiom,(
class_Rings_Ocomm__semiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Ozero__neq__one,axiom,(
class_Rings_Ozero__neq__one(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Orderings_Opreorder,axiom,(
class_Orderings_Opreorder(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Orderings_Olinorder,axiom,(
class_Orderings_Olinorder(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Omonoid__mult,axiom,(
class_Groups_Omonoid__mult(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Omonoid__add,axiom,(
class_Groups_Omonoid__add(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Osemiring__1,axiom,(
class_Rings_Osemiring__1(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Osemiring__0,axiom,(
class_Rings_Osemiring__0(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Omult__zero,axiom,(
class_Rings_Omult__zero(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Orderings_Oorder,axiom,(
class_Orderings_Oorder(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Rings_Osemiring,axiom,(
class_Rings_Osemiring(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Orderings_Oord,axiom,(
class_Orderings_Oord(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Power_Opower,axiom,(
class_Power_Opower(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Ozero,axiom,(
class_Groups_Ozero(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Int_Onumber,axiom,(
class_Int_Onumber(tc_Nat_Onat) )).
fof(arity_Nat__Onat__Groups_Oone,axiom,(
class_Groups_Oone(tc_Nat_Onat) )).
fof(arity_HOL__Obool__Orderings_Opreorder,axiom,(
class_Orderings_Opreorder(tc_HOL_Obool) )).
fof(arity_HOL__Obool__Orderings_Oorder,axiom,(
class_Orderings_Oorder(tc_HOL_Obool) )).
fof(arity_HOL__Obool__Orderings_Oord,axiom,(
class_Orderings_Oord(tc_HOL_Obool) )).
fof(arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,(
class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add,axiom,(
class_Groups_Oordered__cancel__ab__semigroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add__imp__le,axiom,(
class_Groups_Oordered__ab__semigroup__add__imp__le(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict,axiom,(
class_Rings_Olinordered__comm__semiring__strict(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Fields_Olinordered__field__inverse__zero,axiom,(
class_Fields_Olinordered__field__inverse__zero(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict,axiom,(
class_Rings_Olinordered__semiring__1__strict(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra,axiom,(
class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__semiring__strict,axiom,(
class_Rings_Olinordered__semiring__strict(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Odivision__ring__inverse__zero,axiom,(
class_Rings_Odivision__ring__inverse__zero(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1,axiom,(
class_RealVector_Oreal__normed__algebra__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add,axiom,(
class_Groups_Oordered__ab__semigroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs,axiom,(
class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra,axiom,(
class_RealVector_Oreal__normed__algebra(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add,axiom,(
class_Groups_Oordered__comm__monoid__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Olinordered__ab__group__add,axiom,(
class_Groups_Olinordered__ab__group__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add,axiom,(
class_Groups_Ocancel__ab__semigroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors,axiom,(
class_Rings_Oring__1__no__zero__divisors(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oordered__cancel__semiring,axiom,(
class_Rings_Oordered__cancel__semiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__normed__vector,axiom,(
class_RealVector_Oreal__normed__vector(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__ring__strict,axiom,(
class_Rings_Olinordered__ring__strict(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__normed__field,axiom,(
class_RealVector_Oreal__normed__field(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oring__no__zero__divisors,axiom,(
class_Rings_Oring__no__zero__divisors(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oordered__comm__semiring,axiom,(
class_Rings_Oordered__comm__semiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__semiring__1,axiom,(
class_Rings_Olinordered__semiring__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oordered__ab__group__add,axiom,(
class_Groups_Oordered__ab__group__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ocancel__semigroup__add,axiom,(
class_Groups_Ocancel__semigroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__semiring,axiom,(
class_Rings_Olinordered__semiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Fields_Ofield__inverse__zero,axiom,(
class_Fields_Ofield__inverse__zero(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__semidom,axiom,(
class_Rings_Olinordered__semidom(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oab__semigroup__mult,axiom,(
class_Groups_Oab__semigroup__mult(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ocomm__monoid__mult,axiom,(
class_Groups_Ocomm__monoid__mult(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oab__semigroup__add,axiom,(
class_Groups_Oab__semigroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Fields_Olinordered__field,axiom,(
class_Fields_Olinordered__field(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oordered__semiring,axiom,(
class_Rings_Oordered__semiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oordered__ring__abs,axiom,(
class_Rings_Oordered__ring__abs(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Ono__zero__divisors,axiom,(
class_Rings_Ono__zero__divisors(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ocomm__monoid__add,axiom,(
class_Groups_Ocomm__monoid__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__ring,axiom,(
class_Rings_Olinordered__ring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Olinordered__idom,axiom,(
class_Rings_Olinordered__idom(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Ocomm__semiring__1,axiom,(
class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__RealVector_Oreal__field,axiom,(
class_RealVector_Oreal__field(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Odivision__ring,axiom,(
class_Rings_Odivision__ring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Ocomm__semiring,axiom,(
class_Rings_Ocomm__semiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oab__group__add,axiom,(
class_Groups_Oab__group__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Ozero__neq__one,axiom,(
class_Rings_Ozero__neq__one(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oordered__ring,axiom,(
class_Rings_Oordered__ring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Orderings_Opreorder,axiom,(
class_Orderings_Opreorder(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Orderings_Olinorder,axiom,(
class_Orderings_Olinorder(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Omonoid__mult,axiom,(
class_Groups_Omonoid__mult(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Ocomm__ring__1,axiom,(
class_Rings_Ocomm__ring__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Omonoid__add,axiom,(
class_Groups_Omonoid__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Osemiring__1,axiom,(
class_Rings_Osemiring__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Osemiring__0,axiom,(
class_Rings_Osemiring__0(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ogroup__add,axiom,(
class_Groups_Ogroup__add(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Omult__zero,axiom,(
class_Rings_Omult__zero(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Orderings_Oorder,axiom,(
class_Orderings_Oorder(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Int_Oring__char__0,axiom,(
class_Int_Oring__char__0(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Int_Onumber__ring,axiom,(
class_Int_Onumber__ring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Osemiring,axiom,(
class_Rings_Osemiring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Orderings_Oord,axiom,(
class_Orderings_Oord(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oring__1,axiom,(
class_Rings_Oring__1(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Fields_Ofield,axiom,(
class_Fields_Ofield(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Power_Opower,axiom,(
class_Power_Opower(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Ozero,axiom,(
class_Groups_Ozero(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Rings_Oring,axiom,(
class_Rings_Oring(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Int_Onumber,axiom,(
class_Int_Onumber(tc_RealDef_Oreal) )).
fof(arity_RealDef__Oreal__Groups_Oone,axiom,(
class_Groups_Oone(tc_RealDef_Oreal) )).
fof(arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,(
class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra,axiom,(
class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Odivision__ring__inverse__zero,axiom,(
class_Rings_Odivision__ring__inverse__zero(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1,axiom,(
class_RealVector_Oreal__normed__algebra__1(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra,axiom,(
class_RealVector_Oreal__normed__algebra(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add,axiom,(
class_Groups_Ocancel__ab__semigroup__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors,axiom,(
class_Rings_Oring__1__no__zero__divisors(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__vector,axiom,(
class_RealVector_Oreal__normed__vector(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__field,axiom,(
class_RealVector_Oreal__normed__field(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors,axiom,(
class_Rings_Oring__no__zero__divisors(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add,axiom,(
class_Groups_Ocancel__semigroup__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Fields_Ofield__inverse__zero,axiom,(
class_Fields_Ofield__inverse__zero(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Oab__semigroup__mult,axiom,(
class_Groups_Oab__semigroup__mult(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult,axiom,(
class_Groups_Ocomm__monoid__mult(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Oab__semigroup__add,axiom,(
class_Groups_Oab__semigroup__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Ono__zero__divisors,axiom,(
class_Rings_Ono__zero__divisors(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ocomm__monoid__add,axiom,(
class_Groups_Ocomm__monoid__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Ocomm__semiring__1,axiom,(
class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__RealVector_Oreal__field,axiom,(
class_RealVector_Oreal__field(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Odivision__ring,axiom,(
class_Rings_Odivision__ring(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Ocomm__semiring,axiom,(
class_Rings_Ocomm__semiring(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Oab__group__add,axiom,(
class_Groups_Oab__group__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Ozero__neq__one,axiom,(
class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Omonoid__mult,axiom,(
class_Groups_Omonoid__mult(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Ocomm__ring__1,axiom,(
class_Rings_Ocomm__ring__1(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Omonoid__add,axiom,(
class_Groups_Omonoid__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Osemiring__1,axiom,(
class_Rings_Osemiring__1(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Osemiring__0,axiom,(
class_Rings_Osemiring__0(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ogroup__add,axiom,(
class_Groups_Ogroup__add(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Omult__zero,axiom,(
class_Rings_Omult__zero(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Int_Oring__char__0,axiom,(
class_Int_Oring__char__0(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Int_Onumber__ring,axiom,(
class_Int_Onumber__ring(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Osemiring,axiom,(
class_Rings_Osemiring(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Oring__1,axiom,(
class_Rings_Oring__1(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Fields_Ofield,axiom,(
class_Fields_Ofield(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Power_Opower,axiom,(
class_Power_Opower(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Ozero,axiom,(
class_Groups_Ozero(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Rings_Oring,axiom,(
class_Rings_Oring(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Int_Onumber,axiom,(
class_Int_Onumber(tc_Complex_Ocomplex) )).
fof(arity_Complex__Ocomplex__Groups_Oone,axiom,(
class_Groups_Oone(tc_Complex_Ocomplex) )).
%----Conjectures (1)
fof(conj_0,conjecture,(
c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Int_Onumber__class_Onumber__of(tc_RealDef_Oreal,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),v_x____))),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Int_Onumber__class_Onumber__of(tc_Nat_Onat,c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).
%------------------------------------------------------------------------------
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