SET007 Axioms: SET007+774.ax
%------------------------------------------------------------------------------
% File : SET007+774 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Inner Products and Angles of Complex Numbers
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : complex2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 113 ( 9 unt; 0 def)
% Number of atoms : 495 ( 196 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 497 ( 115 ~; 21 |; 99 &)
% ( 13 <=>; 249 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 47 ( 47 usr; 12 con; 0-4 aty)
% Number of variables : 223 ( 223 !; 0 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_complex2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_xcmplx_0(k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1))) = k2_xcmplx_0(k1_real_1(A),k3_xcmplx_0(k1_real_1(B),k7_complex1)) ) ) ).
fof(t2_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( v1_xreal_0(C)
=> ~ ( C = k2_xcmplx_0(k3_xcmplx_0(B,k4_xcmplx_0(k1_int_1(k7_xcmplx_0(A,B)))),A)
& r1_xreal_0(np__0,C)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t3_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(np__0,B)
& r1_xreal_0(np__0,C) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(A,B)
| r1_xreal_0(A,C)
| ! [D] :
( v1_int_1(D)
=> ( B = k2_xcmplx_0(C,k3_xcmplx_0(A,D))
=> B = C ) ) ) ) ) ) ) ).
fof(t4_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k19_sin_cos(k6_xcmplx_0(A,B)) = k6_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(B)),k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B)))
& k22_sin_cos(k6_xcmplx_0(A,B)) = k2_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B))) ) ) ) ).
fof(t5_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_xcmplx_0(A,k32_sin_cos)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_xcmplx_0(A,k32_sin_cos)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ).
fof(t6_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k19_sin_cos(k6_xcmplx_0(A,k32_sin_cos)) = k4_xcmplx_0(k19_sin_cos(A))
& k22_sin_cos(k6_xcmplx_0(A,k32_sin_cos)) = k4_xcmplx_0(k22_sin_cos(A)) ) ) ).
fof(t7_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r2_hidden(A,k2_rcomp_1(np__0,k6_real_1(k32_sin_cos,np__2)))
& r2_hidden(B,k2_rcomp_1(np__0,k6_real_1(k32_sin_cos,np__2))) )
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k19_sin_cos(B),k19_sin_cos(A)) )
& ~ ( ~ r1_xreal_0(k19_sin_cos(B),k19_sin_cos(A))
& r1_xreal_0(B,A) ) ) ) ) ) ).
fof(t8_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r2_hidden(A,k2_rcomp_1(k6_real_1(k32_sin_cos,np__2),k32_sin_cos))
& r2_hidden(B,k2_rcomp_1(k6_real_1(k32_sin_cos,np__2),k32_sin_cos)) )
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k19_sin_cos(A),k19_sin_cos(B)) )
& ~ ( ~ r1_xreal_0(k19_sin_cos(A),k19_sin_cos(B))
& r1_xreal_0(B,A) ) ) ) ) ) ).
fof(t9_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> k19_sin_cos(A) = k19_sin_cos(k2_xcmplx_0(k3_xcmplx_0(k4_real_1(np__2,k32_sin_cos),B),A)) ) ) ).
fof(t10_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> k22_sin_cos(A) = k22_sin_cos(k2_xcmplx_0(k3_xcmplx_0(k4_real_1(np__2,k32_sin_cos),B),A)) ) ) ).
fof(t11_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( k19_sin_cos(A) = np__0
& k22_sin_cos(A) = np__0 ) ) ).
fof(t12_complex2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B)
& k19_sin_cos(A) = k19_sin_cos(B)
& k22_sin_cos(A) = k22_sin_cos(B) )
=> ( r1_xreal_0(k4_real_1(np__2,k32_sin_cos),A)
| r1_xreal_0(k4_real_1(np__2,k32_sin_cos),B)
| A = B ) ) ) ) ).
fof(d1_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k1_complex2(A) = A ) ).
fof(t13_complex2,axiom,
$true ).
fof(t14_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k1_complex2(k8_complex1(A,B)) = k2_xcmplx_0(k1_complex2(A),k1_complex2(B)) ) ) ).
fof(t15_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( A = np__0
<=> k1_complex2(A) = k1_rlvect_1(k1_complfld) ) ) ).
fof(t16_complex2,axiom,
$true ).
fof(t17_complex2,axiom,
$true ).
fof(t18_complex2,axiom,
$true ).
fof(t19_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> A = k5_arytm_0(k4_real_1(k17_complex1(A),k23_sin_cos(k1_comptrig(A))),k4_real_1(k17_complex1(A),k20_sin_cos(k1_comptrig(A)))) ) ).
fof(t20_complex2,axiom,
k1_comptrig(np__0) = np__0 ).
fof(t21_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( A = k5_arytm_0(k4_real_1(k17_complex1(A),k23_sin_cos(B)),k4_real_1(k17_complex1(A),k20_sin_cos(B)))
& r1_xreal_0(np__0,B) )
=> ( A = np__0
| r1_xreal_0(k4_real_1(np__2,k32_sin_cos),B)
| B = k1_comptrig(A) ) ) ) ) ).
fof(t22_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( A != np__0
=> ( ( ~ r1_xreal_0(k32_sin_cos,k1_comptrig(A))
=> k1_comptrig(k4_xcmplx_0(A)) = k3_real_1(k1_comptrig(A),k32_sin_cos) )
& ( r1_xreal_0(k32_sin_cos,k1_comptrig(A))
=> k1_comptrig(k4_xcmplx_0(A)) = k5_real_1(k1_comptrig(A),k32_sin_cos) ) ) ) ) ).
fof(t23_complex2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__0,A)
=> k1_comptrig(k5_arytm_0(A,np__0)) = np__0 ) ) ).
fof(t24_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k1_comptrig(A) = np__0
<=> A = k5_arytm_0(k17_complex1(A),np__0) ) ) ).
fof(t25_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( A != np__0
=> ( ~ r1_xreal_0(k32_sin_cos,k1_comptrig(A))
<=> r1_xreal_0(k32_sin_cos,k1_comptrig(k4_xcmplx_0(A))) ) ) ) ).
fof(t26_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ~ ( A = B
& k6_xcmplx_0(A,B) = np__0 )
=> ( ~ r1_xreal_0(k32_sin_cos,k1_comptrig(k6_xcmplx_0(A,B)))
<=> r1_xreal_0(k32_sin_cos,k1_comptrig(k6_xcmplx_0(B,A))) ) ) ) ) ).
fof(t27_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r2_hidden(k1_comptrig(A),k2_rcomp_1(np__0,k32_sin_cos))
<=> ~ r1_xreal_0(k4_complex1(A),np__0) ) ) ).
fof(t28_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k1_comptrig(A) != np__0
=> ( ~ ( ~ r1_xreal_0(k32_sin_cos,k1_comptrig(A))
& r1_xreal_0(k20_sin_cos(k1_comptrig(A)),np__0) )
& ~ ( ~ r1_xreal_0(k20_sin_cos(k1_comptrig(A)),np__0)
& r1_xreal_0(k32_sin_cos,k1_comptrig(A)) ) ) ) ) ).
fof(t29_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ~ ( ~ r1_xreal_0(k32_sin_cos,k1_comptrig(A))
& ~ r1_xreal_0(k32_sin_cos,k1_comptrig(B))
& r1_xreal_0(k32_sin_cos,k1_comptrig(k2_xcmplx_0(A,B))) ) ) ) ).
fof(t30_complex2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_comptrig(k5_arytm_0(np__0,A)) = k6_real_1(k32_sin_cos,np__2) ) ) ).
fof(t31_complex2,axiom,
$true ).
fof(t32_complex2,axiom,
$true ).
fof(t33_complex2,axiom,
$true ).
fof(t34_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k1_comptrig(A) = np__0
<=> ( r1_xreal_0(np__0,k3_complex1(A))
& k4_complex1(A) = np__0 ) ) ) ).
fof(t35_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k1_comptrig(A) = k32_sin_cos
<=> ( ~ r1_xreal_0(np__0,k3_complex1(A))
& k4_complex1(A) = np__0 ) ) ) ).
fof(t36_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k4_complex1(A) = np__0
<=> ( k1_comptrig(A) = np__0
| k1_comptrig(A) = k32_sin_cos ) ) ) ).
fof(t37_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(k1_comptrig(A),k32_sin_cos)
=> r1_xreal_0(np__0,k4_complex1(A)) ) ) ).
fof(t38_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( A != np__0
=> ( k23_sin_cos(k1_comptrig(k10_complex1(A))) = k1_real_1(k23_sin_cos(k1_comptrig(A)))
& k20_sin_cos(k1_comptrig(k10_complex1(A))) = k1_real_1(k20_sin_cos(k1_comptrig(A))) ) ) ) ).
fof(t39_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( A != np__0
=> ( k23_sin_cos(k1_comptrig(A)) = k6_real_1(k3_complex1(A),k17_complex1(A))
& k20_sin_cos(k1_comptrig(A)) = k6_real_1(k4_complex1(A),k17_complex1(A)) ) ) ) ).
fof(t40_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k1_comptrig(k3_xcmplx_0(A,k5_arytm_0(B,np__0))) = k1_comptrig(A) ) ) ) ).
fof(t41_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(np__0,B)
=> k1_comptrig(k3_xcmplx_0(A,k5_arytm_0(B,np__0))) = k1_comptrig(k4_xcmplx_0(A)) ) ) ) ).
fof(d2_complex2,axiom,
$true ).
fof(d3_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k2_complex2(A,B) = k3_xcmplx_0(A,k15_complex1(B)) ) ) ).
fof(t42_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(A,B) = k5_arytm_0(k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B))),k3_real_1(k1_real_1(k4_real_1(k3_complex1(A),k4_complex1(B))),k4_real_1(k4_complex1(A),k3_complex1(B)))) ) ) ).
fof(t43_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k2_complex2(A,A) = k5_arytm_0(k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(A)),k4_real_1(k4_complex1(A),k4_complex1(A))),np__0)
& k2_complex2(A,A) = k5_arytm_0(k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A))),np__0) ) ) ).
fof(t44_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k2_complex2(A,A) = k5_arytm_0(k7_square_1(k17_complex1(A)),np__0)
& k7_square_1(k17_complex1(A)) = k3_complex1(k2_complex2(A,A)) ) ) ).
fof(t45_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k17_complex1(k2_complex2(A,B)) = k4_real_1(k17_complex1(A),k17_complex1(B)) ) ) ).
fof(t46_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k2_complex2(A,A) = np__0
=> A = np__0 ) ) ).
fof(t47_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(A,B) = k15_complex1(k2_complex2(B,A)) ) ) ).
fof(t48_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(k8_complex1(A,B),C) = k8_complex1(k2_complex2(A,C),k2_complex2(B,C)) ) ) ) ).
fof(t49_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(A,k8_complex1(B,C)) = k8_complex1(k2_complex2(A,B),k2_complex2(A,C)) ) ) ) ).
fof(t50_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(k9_complex1(A,B),C) = k9_complex1(A,k2_complex2(B,C)) ) ) ) ).
fof(t51_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(A,k9_complex1(B,C)) = k9_complex1(k15_complex1(B),k2_complex2(A,C)) ) ) ) ).
fof(t52_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(k9_complex1(A,B),C) = k2_complex2(B,k9_complex1(k15_complex1(A),C)) ) ) ) ).
fof(t53_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> ! [E] :
( m1_subset_1(E,k2_numbers)
=> k2_complex2(k8_complex1(k9_complex1(A,B),k9_complex1(C,D)),E) = k8_complex1(k9_complex1(A,k2_complex2(B,E)),k9_complex1(C,k2_complex2(D,E))) ) ) ) ) ) ).
fof(t54_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> ! [E] :
( m1_subset_1(E,k2_numbers)
=> k2_complex2(A,k8_complex1(k9_complex1(B,C),k9_complex1(D,E))) = k8_complex1(k9_complex1(k15_complex1(B),k2_complex2(A,C)),k9_complex1(k15_complex1(D),k2_complex2(A,E))) ) ) ) ) ) ).
fof(t55_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(k10_complex1(A),B) = k2_complex2(A,k10_complex1(B)) ) ) ).
fof(t56_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(k10_complex1(A),B) = k10_complex1(k2_complex2(A,B)) ) ) ).
fof(t57_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k10_complex1(k2_complex2(A,B)) = k2_complex2(A,k10_complex1(B)) ) ) ).
fof(t58_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(k10_complex1(A),k10_complex1(B)) = k2_complex2(A,B) ) ) ).
fof(t59_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(k11_complex1(A,B),C) = k11_complex1(k2_complex2(A,C),k2_complex2(B,C)) ) ) ) ).
fof(t60_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k2_complex2(A,k11_complex1(B,C)) = k11_complex1(k2_complex2(A,B),k2_complex2(A,C)) ) ) ) ).
fof(t61_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k2_complex2(np__0,A) = k5_complex1
& k2_complex2(A,np__0) = np__0 ) ) ).
fof(t62_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(k8_complex1(A,B),k8_complex1(A,B)) = k8_complex1(k8_complex1(k8_complex1(k2_complex2(A,A),k2_complex2(A,B)),k2_complex2(B,A)),k2_complex2(B,B)) ) ) ).
fof(t63_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_complex2(k11_complex1(A,B),k11_complex1(A,B)) = k8_complex1(k11_complex1(k11_complex1(k2_complex2(A,A),k2_complex2(A,B)),k2_complex2(B,A)),k2_complex2(B,B)) ) ) ).
fof(t64_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k3_complex1(k2_complex2(A,B)) = np__0
<=> ( k4_complex1(k2_complex2(A,B)) = k4_real_1(k17_complex1(A),k17_complex1(B))
| k4_complex1(k2_complex2(A,B)) = k1_real_1(k4_real_1(k17_complex1(A),k17_complex1(B))) ) ) ) ) ).
fof(d4_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k3_complex2(A,B) = k5_arytm_0(k4_real_1(k17_complex1(A),k23_sin_cos(k3_real_1(B,k1_comptrig(A)))),k4_real_1(k17_complex1(A),k20_sin_cos(k3_real_1(B,k1_comptrig(A))))) ) ) ).
fof(t65_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k3_complex2(A,np__0) = A ) ).
fof(t66_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_complex2(A,B) = np__0
<=> A = np__0 ) ) ) ).
fof(t67_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k17_complex1(k3_complex2(A,B)) = k17_complex1(A) ) ) ).
fof(t68_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( A != np__0
& ! [C] :
( v1_int_1(C)
=> k1_comptrig(k3_complex2(A,B)) != k2_xcmplx_0(k3_xcmplx_0(k4_real_1(np__2,k32_sin_cos),C),k3_real_1(B,k1_comptrig(A))) ) ) ) ) ).
fof(t69_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k3_complex2(A,k1_real_1(k1_comptrig(A))) = k5_arytm_0(k17_complex1(A),np__0) ) ).
fof(t70_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_complex1(k3_complex2(A,B)) = k5_real_1(k4_real_1(k3_complex1(A),k23_sin_cos(B)),k4_real_1(k4_complex1(A),k20_sin_cos(B)))
& k4_complex1(k3_complex2(A,B)) = k3_real_1(k4_real_1(k3_complex1(A),k20_sin_cos(B)),k4_real_1(k4_complex1(A),k23_sin_cos(B))) ) ) ) ).
fof(t71_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k3_complex2(k8_complex1(A,B),C) = k8_complex1(k3_complex2(A,C),k3_complex2(B,C)) ) ) ) ).
fof(t72_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k3_complex2(k10_complex1(A),B) = k10_complex1(k3_complex2(A,B)) ) ) ).
fof(t73_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k3_complex2(k11_complex1(A,B),C) = k11_complex1(k3_complex2(A,C),k3_complex2(B,C)) ) ) ) ).
fof(t74_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k3_complex2(A,k32_sin_cos) = k10_complex1(A) ) ).
fof(d5_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( ( ~ ( k1_comptrig(A) != np__0
& B = np__0 )
=> k4_complex2(A,B) = k1_comptrig(k3_complex2(B,k1_real_1(k1_comptrig(A)))) )
& ~ ( k1_comptrig(A) != np__0
& B = np__0
& k4_complex2(A,B) != k5_real_1(k4_real_1(np__2,k32_sin_cos),k1_comptrig(A)) ) ) ) ) ).
fof(t75_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(np__0,B)
=> k4_complex2(k5_arytm_0(B,np__0),A) = k1_comptrig(A) ) ) ) ).
fof(t76_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( k1_comptrig(A) = k1_comptrig(B)
=> ( A = np__0
| B = np__0
| k1_comptrig(k3_complex2(A,C)) = k1_comptrig(k3_complex2(B,C)) ) ) ) ) ) ).
fof(t77_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k4_complex2(A,B) = k4_complex2(k9_complex1(A,k5_arytm_0(C,np__0)),k9_complex1(B,k5_arytm_0(C,np__0))) ) ) ) ) ).
fof(t78_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k1_comptrig(A) = k1_comptrig(B)
=> ( A = np__0
| B = np__0
| k1_comptrig(k10_complex1(A)) = k1_comptrig(k10_complex1(B)) ) ) ) ) ).
fof(t79_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( A != np__0
& B != np__0
& k4_complex2(A,B) != k4_complex2(k3_complex2(A,C),k3_complex2(B,C)) ) ) ) ) ).
fof(t80_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(np__0,C)
& A != np__0
& B != np__0
& k4_complex2(A,B) != k4_complex2(k9_complex1(A,k5_arytm_0(C,np__0)),k9_complex1(B,k5_arytm_0(C,np__0))) ) ) ) ) ).
fof(t81_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ~ ( A != np__0
& B != np__0
& k4_complex2(A,B) != k4_complex2(k10_complex1(A),k10_complex1(B)) ) ) ) ).
fof(t82_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k4_complex2(B,A) = np__0
=> ( A = np__0
| k4_complex2(B,k10_complex1(A)) = k32_sin_cos ) ) ) ) ).
fof(t83_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ~ ( A != np__0
& B != np__0
& ~ ( k23_sin_cos(k4_complex2(A,B)) = k6_real_1(k3_complex1(k2_complex2(A,B)),k4_real_1(k17_complex1(A),k17_complex1(B)))
& k20_sin_cos(k4_complex2(A,B)) = k1_real_1(k6_real_1(k4_complex1(k2_complex2(A,B)),k4_real_1(k17_complex1(A),k17_complex1(B)))) ) ) ) ) ).
fof(d6_complex2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( v1_xcmplx_0(C)
=> ( ( r1_xreal_0(np__0,k5_real_1(k1_comptrig(k6_xcmplx_0(C,B)),k1_comptrig(k6_xcmplx_0(A,B))))
=> k5_complex2(A,B,C) = k5_real_1(k1_comptrig(k6_xcmplx_0(C,B)),k1_comptrig(k6_xcmplx_0(A,B))) )
& ( ~ r1_xreal_0(np__0,k5_real_1(k1_comptrig(k6_xcmplx_0(C,B)),k1_comptrig(k6_xcmplx_0(A,B))))
=> k5_complex2(A,B,C) = k3_real_1(k4_real_1(np__2,k32_sin_cos),k5_real_1(k1_comptrig(k6_xcmplx_0(C,B)),k1_comptrig(k6_xcmplx_0(A,B)))) ) ) ) ) ) ).
fof(t84_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( r1_xreal_0(np__0,k5_complex2(A,B,C))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k5_complex2(A,B,C)) ) ) ) ) ).
fof(t85_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> k5_complex2(A,B,C) = k5_complex2(k11_complex1(A,B),np__0,k11_complex1(C,B)) ) ) ) ).
fof(t86_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> k5_complex2(A,B,C) = k5_complex2(k8_complex1(A,D),k8_complex1(B,D),k8_complex1(C,D)) ) ) ) ) ).
fof(t87_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_complex2(A,B) = k5_complex2(A,np__0,B) ) ) ).
fof(t88_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k5_complex2(A,B,C) = np__0
=> ( k1_comptrig(k11_complex1(A,B)) = k1_comptrig(k11_complex1(C,B))
& k5_complex2(C,B,A) = np__0 ) ) ) ) ) ).
fof(t89_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ~ ( A != np__0
& B != np__0
& ~ ( k3_complex1(k2_complex2(A,B)) = np__0
<=> ( k5_complex2(A,np__0,B) = k6_real_1(k32_sin_cos,np__2)
| k5_complex2(A,np__0,B) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) ) ) ) ) ) ).
fof(t90_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ~ ( A != np__0
& B != np__0
& ~ ( ~ ( ( k4_complex1(k2_complex2(A,B)) = k4_real_1(k17_complex1(A),k17_complex1(B))
| k4_complex1(k2_complex2(A,B)) = k1_real_1(k4_real_1(k17_complex1(A),k17_complex1(B))) )
& k5_complex2(A,np__0,B) != k6_real_1(k32_sin_cos,np__2)
& k5_complex2(A,np__0,B) != k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& ~ ( ( k5_complex2(A,np__0,B) = k6_real_1(k32_sin_cos,np__2)
| k5_complex2(A,np__0,B) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& k4_complex1(k2_complex2(A,B)) != k4_real_1(k17_complex1(A),k17_complex1(B))
& k4_complex1(k2_complex2(A,B)) != k1_real_1(k4_real_1(k17_complex1(A),k17_complex1(B))) ) ) ) ) ) ).
fof(t91_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( A != B
& C != B
& ( k5_complex2(A,B,C) = k6_real_1(k32_sin_cos,np__2)
| k5_complex2(A,B,C) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) )
& k3_real_1(k7_square_1(k17_complex1(k11_complex1(A,B))),k7_square_1(k17_complex1(k11_complex1(C,B)))) != k7_square_1(k17_complex1(k11_complex1(A,C))) ) ) ) ) ).
fof(t92_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( A != B
& B != C
& k5_complex2(A,B,C) != k5_complex2(k3_complex2(A,D),k3_complex2(B,D),k3_complex2(C,D)) ) ) ) ) ) ).
fof(t93_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k5_complex2(A,B,A) = np__0 ) ) ).
fof(t94_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k5_complex2(A,B,C) != np__0
<=> k2_xcmplx_0(k5_complex2(A,B,C),k5_complex2(C,B,A)) = k4_real_1(np__2,k32_sin_cos) ) ) ) ) ).
fof(t95_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( k5_complex2(A,B,C) != np__0
& k5_complex2(C,B,A) = np__0 ) ) ) ) ).
fof(t96_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k5_complex2(A,B,C) = k32_sin_cos
=> k5_complex2(C,B,A) = k32_sin_cos ) ) ) ) ).
fof(t97_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( A != B
& A != C
& B != C
& k5_complex2(A,B,C) = np__0
& k5_complex2(B,C,A) = np__0
& k5_complex2(C,A,B) = np__0 ) ) ) ) ).
fof(t98_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( A != B
& B != C
& ~ r1_xreal_0(k5_complex2(A,B,C),np__0)
& ~ r1_xreal_0(k32_sin_cos,k5_complex2(A,B,C))
& ~ ( k2_xcmplx_0(k2_xcmplx_0(k5_complex2(A,B,C),k5_complex2(B,C,A)),k5_complex2(C,A,B)) = k32_sin_cos
& ~ r1_xreal_0(k5_complex2(B,C,A),np__0)
& ~ r1_xreal_0(k5_complex2(C,A,B),np__0) ) ) ) ) ) ).
fof(t99_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( A != B
& B != C
& ~ r1_xreal_0(k5_complex2(A,B,C),k32_sin_cos)
& ~ ( k2_xcmplx_0(k2_xcmplx_0(k5_complex2(A,B,C),k5_complex2(B,C,A)),k5_complex2(C,A,B)) = k4_real_1(np__5,k32_sin_cos)
& ~ r1_xreal_0(k5_complex2(B,C,A),k32_sin_cos)
& ~ r1_xreal_0(k5_complex2(C,A,B),k32_sin_cos) ) ) ) ) ) ).
fof(t100_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k5_complex2(A,B,C) = k32_sin_cos
=> ( A = B
| B = C
| ( k5_complex2(B,C,A) = np__0
& k5_complex2(C,A,B) = np__0 ) ) ) ) ) ) ).
fof(t101_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( A != B
& A != C
& B != C
& k5_complex2(A,B,C) = np__0
& ~ ( k5_complex2(B,C,A) = np__0
& k5_complex2(C,A,B) = k32_sin_cos )
& ~ ( k5_complex2(B,C,A) = k32_sin_cos
& k5_complex2(C,A,B) = np__0 ) ) ) ) ) ).
fof(t102_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( ( k2_xcmplx_0(k2_xcmplx_0(k5_complex2(A,B,C),k5_complex2(B,C,A)),k5_complex2(C,A,B)) = k32_sin_cos
| k2_xcmplx_0(k2_xcmplx_0(k5_complex2(A,B,C),k5_complex2(B,C,A)),k5_complex2(C,A,B)) = k4_real_1(np__5,k32_sin_cos) )
<=> ( A != B
& A != C
& B != C ) ) ) ) ) ).
fof(dt_k1_complex2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k1_complex2(A),u1_struct_0(k1_complfld)) ) ).
fof(dt_k2_complex2,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B) )
=> m1_subset_1(k2_complex2(A,B),k2_numbers) ) ).
fof(dt_k3_complex2,axiom,
! [A,B] :
( ( v1_xcmplx_0(A)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k3_complex2(A,B),k2_numbers) ) ).
fof(dt_k4_complex2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> m1_subset_1(k4_complex2(A,B),k1_numbers) ) ).
fof(dt_k5_complex2,axiom,
! [A,B,C] :
( ( v1_xcmplx_0(A)
& v1_xcmplx_0(B)
& v1_xcmplx_0(C) )
=> v1_xreal_0(k5_complex2(A,B,C)) ) ).
%------------------------------------------------------------------------------