SET007 Axioms: SET007+641.ax
%------------------------------------------------------------------------------
% File : SET007+641 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Field 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 : complfld [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 123 ( 45 unt; 0 def)
% Number of atoms : 388 ( 158 equ)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 359 ( 94 ~; 9 |; 74 &)
% ( 2 <=>; 180 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-2 aty)
% Number of functors : 39 ( 39 usr; 10 con; 0-3 aty)
% Number of variables : 154 ( 154 !; 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(fc1_complfld,axiom,
( ~ v3_struct_0(k1_complfld)
& v3_vectsp_1(k1_complfld) ) ).
fof(cc1_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> v1_xcmplx_0(A) ) ).
fof(fc2_complfld,axiom,
( ~ v3_struct_0(k1_complfld)
& v2_group_1(k1_complfld)
& v3_vectsp_1(k1_complfld)
& v6_vectsp_1(k1_complfld)
& v8_vectsp_1(k1_complfld) ) ).
fof(fc3_complfld,axiom,
( ~ v3_struct_0(k1_complfld)
& v3_rlvect_1(k1_complfld)
& v4_rlvect_1(k1_complfld)
& v5_rlvect_1(k1_complfld)
& v6_rlvect_1(k1_complfld)
& v2_group_1(k1_complfld)
& v4_group_1(k1_complfld)
& v7_group_1(k1_complfld)
& v3_vectsp_1(k1_complfld)
& v4_vectsp_1(k1_complfld)
& v5_vectsp_1(k1_complfld)
& v6_vectsp_1(k1_complfld)
& v7_vectsp_1(k1_complfld)
& v8_vectsp_1(k1_complfld)
& v9_vectsp_1(k1_complfld)
& ~ v10_vectsp_1(k1_complfld) ) ).
fof(t1_complfld,axiom,
$true ).
fof(t2_complfld,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k2_xcmplx_0(k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1)),k2_xcmplx_0(C,k3_xcmplx_0(D,k7_complex1))) = k2_xcmplx_0(k2_xcmplx_0(A,C),k3_xcmplx_0(k2_xcmplx_0(B,D),k7_complex1)) ) ) ) ) ).
fof(d1_complfld,axiom,
! [A] :
( ( v3_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( A = k1_complfld
<=> ( u1_struct_0(A) = k2_numbers
& u1_rlvect_1(A) = k27_binop_2
& u1_group_1(A) = k29_binop_2
& u1_vectsp_1(A) = k6_complex1
& u2_struct_0(A) = k5_complex1 ) ) ) ).
fof(t3_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( v1_xcmplx_0(C)
=> ! [D] :
( v1_xcmplx_0(D)
=> ( ( A = C
& B = D )
=> k4_rlvect_1(k1_complfld,A,B) = k2_xcmplx_0(C,D) ) ) ) ) ) ).
fof(t4_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( v1_xcmplx_0(B)
=> ( A = B
=> k5_rlvect_1(k1_complfld,A) = k4_xcmplx_0(B) ) ) ) ).
fof(t5_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( v1_xcmplx_0(C)
=> ! [D] :
( v1_xcmplx_0(D)
=> ( ( A = C
& B = D )
=> k6_rlvect_1(k1_complfld,A,B) = k6_xcmplx_0(C,D) ) ) ) ) ) ).
fof(t6_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( v1_xcmplx_0(C)
=> ! [D] :
( v1_xcmplx_0(D)
=> ( ( A = C
& B = D )
=> k10_group_1(k1_complfld,A,B) = k3_xcmplx_0(C,D) ) ) ) ) ) ).
fof(t7_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( v1_xcmplx_0(B)
=> ( A = B
=> ( A = k1_rlvect_1(k1_complfld)
| k4_vectsp_1(k1_complfld,A) = k5_xcmplx_0(B) ) ) ) ) ).
fof(t8_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( v1_xcmplx_0(C)
=> ! [D] :
( v1_xcmplx_0(D)
=> ( ( A = C
& B = D )
=> ( B = k1_rlvect_1(k1_complfld)
| k5_vectsp_1(k1_complfld,A,B) = k7_xcmplx_0(C,D) ) ) ) ) ) ) ).
fof(t9_complfld,axiom,
k1_rlvect_1(k1_complfld) = k5_complex1 ).
fof(t10_complfld,axiom,
k2_group_1(k1_complfld) = k6_complex1 ).
fof(t11_complfld,axiom,
k4_rlvect_1(k1_complfld,k2_group_1(k1_complfld),k2_group_1(k1_complfld)) != k1_rlvect_1(k1_complfld) ).
fof(t12_complfld,axiom,
$true ).
fof(t13_complfld,axiom,
$true ).
fof(t14_complfld,axiom,
$true ).
fof(t15_complfld,axiom,
$true ).
fof(t16_complfld,axiom,
$true ).
fof(t17_complfld,axiom,
$true ).
fof(t18_complfld,axiom,
$true ).
fof(t19_complfld,axiom,
$true ).
fof(t20_complfld,axiom,
$true ).
fof(t21_complfld,axiom,
$true ).
fof(t22_complfld,axiom,
$true ).
fof(t23_complfld,axiom,
$true ).
fof(t24_complfld,axiom,
$true ).
fof(t25_complfld,axiom,
$true ).
fof(t26_complfld,axiom,
$true ).
fof(t27_complfld,axiom,
$true ).
fof(t28_complfld,axiom,
$true ).
fof(t29_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k5_rlvect_1(k1_complfld,A) = k10_group_1(k1_complfld,k5_rlvect_1(k1_complfld,k2_group_1(k1_complfld)),A) ) ).
fof(t30_complfld,axiom,
$true ).
fof(t31_complfld,axiom,
$true ).
fof(t32_complfld,axiom,
$true ).
fof(t33_complfld,axiom,
$true ).
fof(t34_complfld,axiom,
$true ).
fof(t35_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k6_rlvect_1(k1_complfld,A,k5_rlvect_1(k1_complfld,B)) = k4_rlvect_1(k1_complfld,A,B) ) ) ).
fof(t36_complfld,axiom,
$true ).
fof(t37_complfld,axiom,
$true ).
fof(t38_complfld,axiom,
$true ).
fof(t39_complfld,axiom,
$true ).
fof(t40_complfld,axiom,
$true ).
fof(t41_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> A = k6_rlvect_1(k1_complfld,k4_rlvect_1(k1_complfld,A,B),B) ) ) ).
fof(t42_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> A = k4_rlvect_1(k1_complfld,k6_rlvect_1(k1_complfld,A,B),B) ) ) ).
fof(t43_complfld,axiom,
$true ).
fof(t44_complfld,axiom,
$true ).
fof(t45_complfld,axiom,
$true ).
fof(t46_complfld,axiom,
$true ).
fof(t47_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( k4_vectsp_1(k1_complfld,A) = k4_vectsp_1(k1_complfld,B)
=> ( A = k1_rlvect_1(k1_complfld)
| B = k1_rlvect_1(k1_complfld)
| A = B ) ) ) ) ).
fof(t48_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& ( k10_group_1(k1_complfld,B,A) = u1_vectsp_1(k1_complfld)
| k10_group_1(k1_complfld,A,B) = u1_vectsp_1(k1_complfld) )
& B != k4_vectsp_1(k1_complfld,A) ) ) ) ).
fof(t49_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& ( k10_group_1(k1_complfld,B,A) = C
| k10_group_1(k1_complfld,A,B) = C )
& ~ ( B = k10_group_1(k1_complfld,C,k4_vectsp_1(k1_complfld,A))
& B = k10_group_1(k1_complfld,k4_vectsp_1(k1_complfld,A),C) ) ) ) ) ) ).
fof(t50_complfld,axiom,
k4_vectsp_1(k1_complfld,u1_vectsp_1(k1_complfld)) = u1_vectsp_1(k1_complfld) ).
fof(t51_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k4_vectsp_1(k1_complfld,k10_group_1(k1_complfld,A,B)) != k10_group_1(k1_complfld,k4_vectsp_1(k1_complfld,A),k4_vectsp_1(k1_complfld,B)) ) ) ) ).
fof(t52_complfld,axiom,
$true ).
fof(t53_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k4_vectsp_1(k1_complfld,k5_rlvect_1(k1_complfld,A)) = k5_rlvect_1(k1_complfld,k4_vectsp_1(k1_complfld,A)) ) ) ).
fof(t54_complfld,axiom,
$true ).
fof(t55_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k4_rlvect_1(k1_complfld,k4_vectsp_1(k1_complfld,A),k4_vectsp_1(k1_complfld,B)) != k10_group_1(k1_complfld,k4_rlvect_1(k1_complfld,A,B),k4_vectsp_1(k1_complfld,k10_group_1(k1_complfld,A,B))) ) ) ) ).
fof(t56_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k6_rlvect_1(k1_complfld,k4_vectsp_1(k1_complfld,A),k4_vectsp_1(k1_complfld,B)) != k10_group_1(k1_complfld,k6_rlvect_1(k1_complfld,B,A),k4_vectsp_1(k1_complfld,k10_group_1(k1_complfld,A,B))) ) ) ) ).
fof(t57_complfld,axiom,
$true ).
fof(t58_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k4_vectsp_1(k1_complfld,A) = k5_vectsp_1(k1_complfld,u1_vectsp_1(k1_complfld),A) ) ) ).
fof(t59_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k5_vectsp_1(k1_complfld,A,u1_vectsp_1(k1_complfld)) = A ) ).
fof(t60_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k5_vectsp_1(k1_complfld,A,A) = u1_vectsp_1(k1_complfld) ) ) ).
fof(t61_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k5_vectsp_1(k1_complfld,k1_rlvect_1(k1_complfld),A) = k1_rlvect_1(k1_complfld) ) ) ).
fof(t62_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( k5_vectsp_1(k1_complfld,B,A) = k1_rlvect_1(k1_complfld)
=> ( A = k1_rlvect_1(k1_complfld)
| B = k1_rlvect_1(k1_complfld) ) ) ) ) ).
fof(t63_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k10_group_1(k1_complfld,k5_vectsp_1(k1_complfld,C,A),k5_vectsp_1(k1_complfld,D,B)) != k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,C,D),k10_group_1(k1_complfld,A,B)) ) ) ) ) ) ).
fof(t64_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k10_group_1(k1_complfld,B,k5_vectsp_1(k1_complfld,C,A)) = k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,B,C),A) ) ) ) ) ).
fof(t65_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( k5_vectsp_1(k1_complfld,B,A) = u1_vectsp_1(k1_complfld)
=> ( A = k1_rlvect_1(k1_complfld)
| B = A ) ) ) ) ).
fof(t66_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> B = k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,B,A),A) ) ) ) ).
fof(t67_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k4_vectsp_1(k1_complfld,k5_vectsp_1(k1_complfld,A,B)) != k5_vectsp_1(k1_complfld,B,A) ) ) ) ).
fof(t68_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k5_vectsp_1(k1_complfld,k4_vectsp_1(k1_complfld,A),k4_vectsp_1(k1_complfld,B)) != k5_vectsp_1(k1_complfld,B,A) ) ) ) ).
fof(t69_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k5_vectsp_1(k1_complfld,B,k4_vectsp_1(k1_complfld,A)) = k10_group_1(k1_complfld,B,A) ) ) ) ).
fof(t70_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k5_vectsp_1(k1_complfld,k4_vectsp_1(k1_complfld,A),B) != k4_vectsp_1(k1_complfld,k10_group_1(k1_complfld,A,B)) ) ) ) ).
fof(t71_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k10_group_1(k1_complfld,k4_vectsp_1(k1_complfld,A),k5_vectsp_1(k1_complfld,C,B)) != k5_vectsp_1(k1_complfld,C,k10_group_1(k1_complfld,A,B)) ) ) ) ) ).
fof(t72_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& ~ ( k5_vectsp_1(k1_complfld,C,B) = k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,C,A),k10_group_1(k1_complfld,B,A))
& k5_vectsp_1(k1_complfld,C,B) = k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,A,C),k10_group_1(k1_complfld,A,B)) ) ) ) ) ) ).
fof(t73_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k5_vectsp_1(k1_complfld,C,k10_group_1(k1_complfld,A,B)) != k5_vectsp_1(k1_complfld,k5_vectsp_1(k1_complfld,C,A),B) ) ) ) ) ).
fof(t74_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,C,B),A) != k5_vectsp_1(k1_complfld,C,k5_vectsp_1(k1_complfld,A,B)) ) ) ) ) ).
fof(t75_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& C != k1_rlvect_1(k1_complfld)
& k5_vectsp_1(k1_complfld,k5_vectsp_1(k1_complfld,D,A),k5_vectsp_1(k1_complfld,B,C)) != k5_vectsp_1(k1_complfld,k10_group_1(k1_complfld,D,C),k10_group_1(k1_complfld,A,B)) ) ) ) ) ) ).
fof(t76_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k4_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,C,A),k5_vectsp_1(k1_complfld,D,B)) != k5_vectsp_1(k1_complfld,k4_rlvect_1(k1_complfld,k10_group_1(k1_complfld,C,B),k10_group_1(k1_complfld,D,A)),k10_group_1(k1_complfld,A,B)) ) ) ) ) ) ).
fof(t77_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k4_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,B,A),k5_vectsp_1(k1_complfld,C,A)) = k5_vectsp_1(k1_complfld,k4_rlvect_1(k1_complfld,B,C),A) ) ) ) ) ).
fof(t78_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> ( k5_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,B,A)) = k5_vectsp_1(k1_complfld,k5_rlvect_1(k1_complfld,B),A)
& k5_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,B,A)) = k5_vectsp_1(k1_complfld,B,k5_rlvect_1(k1_complfld,A)) ) ) ) ) ).
fof(t79_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k5_vectsp_1(k1_complfld,B,A) = k5_vectsp_1(k1_complfld,k5_rlvect_1(k1_complfld,B),k5_rlvect_1(k1_complfld,A)) ) ) ) ).
fof(t80_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& B != k1_rlvect_1(k1_complfld)
& k6_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,C,A),k5_vectsp_1(k1_complfld,D,B)) != k5_vectsp_1(k1_complfld,k6_rlvect_1(k1_complfld,k10_group_1(k1_complfld,C,B),k10_group_1(k1_complfld,D,A)),k10_group_1(k1_complfld,A,B)) ) ) ) ) ) ).
fof(t81_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> k6_rlvect_1(k1_complfld,k5_vectsp_1(k1_complfld,B,A),k5_vectsp_1(k1_complfld,C,A)) = k5_vectsp_1(k1_complfld,k6_rlvect_1(k1_complfld,B,C),A) ) ) ) ) ).
fof(t82_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( A != k1_rlvect_1(k1_complfld)
& ( k10_group_1(k1_complfld,B,A) = C
| k10_group_1(k1_complfld,A,B) = C )
& B != k5_vectsp_1(k1_complfld,C,A) ) ) ) ) ).
fof(t83_complfld,axiom,
k2_complfld(u2_struct_0(k1_complfld)) = u2_struct_0(k1_complfld) ).
fof(t84_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( k2_complfld(A) = u2_struct_0(k1_complfld)
=> A = u2_struct_0(k1_complfld) ) ) ).
fof(t85_complfld,axiom,
k2_complfld(u1_vectsp_1(k1_complfld)) = u1_vectsp_1(k1_complfld) ).
fof(t86_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k2_complfld(k2_complfld(A)) = A ) ).
fof(t87_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k2_complfld(k4_rlvect_1(k1_complfld,A,B)) = k4_rlvect_1(k1_complfld,k2_complfld(A),k2_complfld(B)) ) ) ).
fof(t88_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k2_complfld(k5_rlvect_1(k1_complfld,A)) = k5_rlvect_1(k1_complfld,k2_complfld(A)) ) ).
fof(t89_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k2_complfld(k6_rlvect_1(k1_complfld,A,B)) = k6_rlvect_1(k1_complfld,k2_complfld(A),k2_complfld(B)) ) ) ).
fof(t90_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k2_complfld(k10_group_1(k1_complfld,A,B)) = k10_group_1(k1_complfld,k2_complfld(A),k2_complfld(B)) ) ) ).
fof(t91_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != u2_struct_0(k1_complfld)
=> k2_complfld(k4_vectsp_1(k1_complfld,A)) = k4_vectsp_1(k1_complfld,k2_complfld(A)) ) ) ).
fof(t92_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != u2_struct_0(k1_complfld)
=> k2_complfld(k5_vectsp_1(k1_complfld,B,A)) = k5_vectsp_1(k1_complfld,k2_complfld(B),k2_complfld(A)) ) ) ) ).
fof(t93_complfld,axiom,
k3_complfld(u2_struct_0(k1_complfld)) = np__0 ).
fof(t94_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( k3_complfld(A) = np__0
=> A = k1_rlvect_1(k1_complfld) ) ) ).
fof(t95_complfld,axiom,
$true ).
fof(t96_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( ~ ( A != k1_rlvect_1(k1_complfld)
& r1_xreal_0(k3_complfld(A),np__0) )
& ~ ( ~ r1_xreal_0(k3_complfld(A),np__0)
& A = k1_rlvect_1(k1_complfld) ) ) ) ).
fof(t97_complfld,axiom,
k3_complfld(u1_vectsp_1(k1_complfld)) = np__1 ).
fof(t98_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k3_complfld(k5_rlvect_1(k1_complfld,A)) = k3_complfld(A) ) ).
fof(t99_complfld,axiom,
$true ).
fof(t100_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> r1_xreal_0(k3_complfld(k4_rlvect_1(k1_complfld,A,B)),k9_binop_2(k3_complfld(A),k3_complfld(B))) ) ) ).
fof(t101_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> r1_xreal_0(k3_complfld(k6_rlvect_1(k1_complfld,A,B)),k9_binop_2(k3_complfld(A),k3_complfld(B))) ) ) ).
fof(t102_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> r1_xreal_0(k10_binop_2(k3_complfld(A),k3_complfld(B)),k3_complfld(k4_rlvect_1(k1_complfld,A,B))) ) ) ).
fof(t103_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> r1_xreal_0(k10_binop_2(k3_complfld(A),k3_complfld(B)),k3_complfld(k6_rlvect_1(k1_complfld,A,B))) ) ) ).
fof(t104_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k3_complfld(k6_rlvect_1(k1_complfld,A,B)) = k3_complfld(k6_rlvect_1(k1_complfld,B,A)) ) ) ).
fof(t105_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( k3_complfld(k6_rlvect_1(k1_complfld,A,B)) = np__0
<=> A = B ) ) ) ).
fof(t106_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( ~ ( A != B
& r1_xreal_0(k3_complfld(k6_rlvect_1(k1_complfld,A,B)),np__0) )
& ~ ( ~ r1_xreal_0(k3_complfld(k6_rlvect_1(k1_complfld,A,B)),np__0)
& A = B ) ) ) ) ).
fof(t107_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> r1_xreal_0(k3_complfld(k6_rlvect_1(k1_complfld,A,B)),k9_binop_2(k3_complfld(k6_rlvect_1(k1_complfld,A,C)),k3_complfld(k6_rlvect_1(k1_complfld,C,B)))) ) ) ) ).
fof(t108_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> r1_xreal_0(k18_complex1(k10_binop_2(k3_complfld(A),k3_complfld(B))),k3_complfld(k6_rlvect_1(k1_complfld,A,B))) ) ) ).
fof(t109_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k3_complfld(k10_group_1(k1_complfld,A,B)) = k11_binop_2(k3_complfld(A),k3_complfld(B)) ) ) ).
fof(t110_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != u2_struct_0(k1_complfld)
=> k3_complfld(k4_vectsp_1(k1_complfld,A)) = k8_binop_2(k3_complfld(A)) ) ) ).
fof(t111_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( A != u2_struct_0(k1_complfld)
=> k12_binop_2(k3_complfld(B),k3_complfld(A)) = k3_complfld(k5_vectsp_1(k1_complfld,B,A)) ) ) ) ).
fof(dt_k1_complfld,axiom,
( v3_vectsp_1(k1_complfld)
& l3_vectsp_1(k1_complfld) ) ).
fof(dt_k2_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> m1_subset_1(k2_complfld(A),u1_struct_0(k1_complfld)) ) ).
fof(involutiveness_k2_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k2_complfld(k2_complfld(A)) = A ) ).
fof(redefinition_k2_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k2_complfld(A) = k14_complex1(A) ) ).
fof(dt_k3_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> m1_subset_1(k3_complfld(A),k1_numbers) ) ).
fof(projectivity_k3_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k3_complfld(k3_complfld(A)) = k3_complfld(A) ) ).
fof(redefinition_k3_complfld,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k3_complfld(A) = k16_complex1(A) ) ).
%------------------------------------------------------------------------------