SET007 Axioms: SET007+52.ax
%------------------------------------------------------------------------------
% File : SET007+52 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The 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 : complex1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 223 ( 104 unt; 0 def)
% Number of atoms : 478 ( 165 equ)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 265 ( 10 ~; 5 |; 69 &)
% ( 8 <=>; 173 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 45 ( 45 usr; 9 con; 0-5 aty)
% Number of variables : 169 ( 165 !; 4 ?)
% 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(cc1_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> v1_xcmplx_0(A) ) ).
fof(fc1_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( v1_xcmplx_0(k1_complex1(A))
& v1_xreal_0(k1_complex1(A)) ) ) ).
fof(fc2_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( v1_xcmplx_0(k2_complex1(A))
& v1_xreal_0(k2_complex1(A)) ) ) ).
fof(fc3_complex1,axiom,
( v1_xboole_0(k5_complex1)
& v1_xcmplx_0(k5_complex1)
& v1_membered(k5_complex1)
& v2_membered(k5_complex1)
& v3_membered(k5_complex1)
& v4_membered(k5_complex1)
& v5_membered(k5_complex1) ) ).
fof(fc4_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( v1_xcmplx_0(k16_complex1(A))
& v1_xreal_0(k16_complex1(A)) ) ) ).
fof(t1_complex1,axiom,
$true ).
fof(t2_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_xcmplx_0(k5_square_1(A),k5_square_1(B)) = np__0
<=> ( A = np__0
& B = np__0 ) ) ) ) ).
fof(d1_complex1,axiom,
$true ).
fof(d2_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( ( r2_hidden(A,k1_numbers)
=> ( B = k1_complex1(A)
<=> B = A ) )
& ( ~ r2_hidden(A,k1_numbers)
=> ( B = k1_complex1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,np__2,k1_numbers)
& m2_relset_1(C,np__2,k1_numbers)
& A = C
& B = k1_funct_1(C,np__0) ) ) ) ) ) ).
fof(d3_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( ( r2_hidden(A,k1_numbers)
=> ( B = k2_complex1(A)
<=> B = np__0 ) )
& ( ~ r2_hidden(A,k1_numbers)
=> ( B = k2_complex1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,np__2,k1_numbers)
& m2_relset_1(C,np__2,k1_numbers)
& A = C
& B = k1_funct_1(C,np__1) ) ) ) ) ) ).
fof(t3_complex1,axiom,
$true ).
fof(t4_complex1,axiom,
$true ).
fof(t5_complex1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,np__2,k1_numbers)
& m2_relset_1(A,np__2,k1_numbers) )
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& A = k5_funct_4(k1_numbers,np__0,np__1,B,C) ) ) ) ).
fof(t6_complex1,axiom,
$true ).
fof(t7_complex1,axiom,
$true ).
fof(t8_complex1,axiom,
$true ).
fof(t9_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ( k3_complex1(A) = k3_complex1(B)
& k4_complex1(A) = k4_complex1(B) )
=> A = B ) ) ) ).
fof(d4_complex1,axiom,
$true ).
fof(d5_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( A = B
<=> ( k3_complex1(A) = k3_complex1(B)
& k4_complex1(A) = k4_complex1(B) ) ) ) ) ).
fof(d6_complex1,axiom,
k5_complex1 = np__0 ).
fof(d7_complex1,axiom,
k6_complex1 = np__1 ).
fof(d8_complex1,axiom,
k7_complex1 = k5_arytm_0(np__0,np__1) ).
fof(t10_complex1,axiom,
$true ).
fof(t11_complex1,axiom,
$true ).
fof(t12_complex1,axiom,
( k3_complex1(np__0) = np__0
& k4_complex1(np__0) = np__0 ) ).
fof(t13_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( A = np__0
<=> k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A))) = np__0 ) ) ).
fof(t14_complex1,axiom,
$true ).
fof(t15_complex1,axiom,
( k3_complex1(k6_complex1) = np__1
& k4_complex1(k6_complex1) = np__0 ) ).
fof(t16_complex1,axiom,
$true ).
fof(t17_complex1,axiom,
( k3_complex1(k7_complex1) = np__0
& k4_complex1(k7_complex1) = np__1 ) ).
fof(d9_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k8_complex1(A,B) = k2_xcmplx_0(k3_real_1(k3_complex1(A),k3_complex1(B)),k3_xcmplx_0(k3_real_1(k4_complex1(A),k4_complex1(B)),k7_complex1)) ) ) ).
fof(t18_complex1,axiom,
$true ).
fof(t19_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( k3_complex1(k2_xcmplx_0(A,B)) = k3_real_1(k3_complex1(A),k3_complex1(B))
& k4_complex1(k2_xcmplx_0(A,B)) = k3_real_1(k4_complex1(A),k4_complex1(B)) ) ) ) ).
fof(d10_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k9_complex1(A,B) = k2_xcmplx_0(k5_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B))),k3_xcmplx_0(k3_real_1(k4_real_1(k3_complex1(A),k4_complex1(B)),k4_real_1(k3_complex1(B),k4_complex1(A))),k7_complex1)) ) ) ).
fof(t20_complex1,axiom,
$true ).
fof(t21_complex1,axiom,
$true ).
fof(t22_complex1,axiom,
$true ).
fof(t23_complex1,axiom,
$true ).
fof(t24_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( k3_complex1(k3_xcmplx_0(A,B)) = k5_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B)))
& k4_complex1(k3_xcmplx_0(A,B)) = k3_real_1(k4_real_1(k3_complex1(A),k4_complex1(B)),k4_real_1(k3_complex1(B),k4_complex1(A))) ) ) ) ).
fof(t25_complex1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k3_complex1(k3_xcmplx_0(A,k7_complex1)) = np__0 ) ).
fof(t26_complex1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_complex1(k3_xcmplx_0(A,k7_complex1)) = A ) ).
fof(t27_complex1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_arytm_0(A,B) = k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1)) ) ) ).
fof(t28_complex1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k3_complex1(k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1))) = A
& k4_complex1(k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1))) = B ) ) ) ).
fof(t29_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k2_xcmplx_0(k3_complex1(A),k3_xcmplx_0(k4_complex1(A),k7_complex1)) = A ) ).
fof(t30_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ( k4_complex1(A) = np__0
& k4_complex1(B) = np__0 )
=> ( k3_complex1(k3_xcmplx_0(A,B)) = k4_real_1(k3_complex1(A),k3_complex1(B))
& k4_complex1(k3_xcmplx_0(A,B)) = np__0 ) ) ) ) ).
fof(t31_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ( k3_complex1(A) = np__0
& k3_complex1(B) = np__0 )
=> ( k3_complex1(k3_xcmplx_0(A,B)) = k1_real_1(k4_real_1(k4_complex1(A),k4_complex1(B)))
& k4_complex1(k3_xcmplx_0(A,B)) = np__0 ) ) ) ) ).
fof(t32_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(k9_complex1(A,A)) = k5_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))
& k4_complex1(k9_complex1(A,A)) = k4_real_1(np__2,k4_real_1(k3_complex1(A),k4_complex1(A))) ) ) ).
fof(d11_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k10_complex1(A) = k2_xcmplx_0(k1_real_1(k3_complex1(A)),k3_xcmplx_0(k1_real_1(k4_complex1(A)),k7_complex1)) ) ).
fof(t33_complex1,axiom,
$true ).
fof(t34_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k3_complex1(k4_xcmplx_0(A)) = k1_real_1(k3_complex1(A))
& k4_complex1(k4_xcmplx_0(A)) = k1_real_1(k4_complex1(A)) ) ) ).
fof(t35_complex1,axiom,
$true ).
fof(t36_complex1,axiom,
$true ).
fof(t37_complex1,axiom,
k9_complex1(k7_complex1,k7_complex1) = k10_complex1(k6_complex1) ).
fof(d12_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k11_complex1(A,B) = k2_xcmplx_0(k5_real_1(k3_complex1(A),k3_complex1(B)),k3_xcmplx_0(k5_real_1(k4_complex1(A),k4_complex1(B)),k7_complex1)) ) ) ).
fof(t38_complex1,axiom,
$true ).
fof(t39_complex1,axiom,
$true ).
fof(t40_complex1,axiom,
$true ).
fof(t41_complex1,axiom,
$true ).
fof(t42_complex1,axiom,
$true ).
fof(t43_complex1,axiom,
$true ).
fof(t44_complex1,axiom,
$true ).
fof(t45_complex1,axiom,
$true ).
fof(t46_complex1,axiom,
$true ).
fof(t47_complex1,axiom,
$true ).
fof(t48_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k3_complex1(k11_complex1(A,B)) = k5_real_1(k3_complex1(A),k3_complex1(B))
& k4_complex1(k11_complex1(A,B)) = k5_real_1(k4_complex1(A),k4_complex1(B)) ) ) ) ).
fof(d13_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k12_complex1(A) = k2_xcmplx_0(k6_real_1(k3_complex1(A),k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))),k3_xcmplx_0(k6_real_1(k1_real_1(k4_complex1(A)),k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))),k7_complex1)) ) ).
fof(t49_complex1,axiom,
$true ).
fof(t50_complex1,axiom,
$true ).
fof(t51_complex1,axiom,
$true ).
fof(t52_complex1,axiom,
$true ).
fof(t53_complex1,axiom,
$true ).
fof(t54_complex1,axiom,
$true ).
fof(t55_complex1,axiom,
$true ).
fof(t56_complex1,axiom,
$true ).
fof(t57_complex1,axiom,
$true ).
fof(t58_complex1,axiom,
$true ).
fof(t59_complex1,axiom,
$true ).
fof(t60_complex1,axiom,
$true ).
fof(t61_complex1,axiom,
$true ).
fof(t62_complex1,axiom,
$true ).
fof(t63_complex1,axiom,
$true ).
fof(t64_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k3_complex1(k5_xcmplx_0(A)) = k6_real_1(k3_complex1(A),k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A))))
& k4_complex1(k5_xcmplx_0(A)) = k6_real_1(k1_real_1(k4_complex1(A)),k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))) ) ) ).
fof(t65_complex1,axiom,
$true ).
fof(t66_complex1,axiom,
$true ).
fof(t67_complex1,axiom,
$true ).
fof(t68_complex1,axiom,
$true ).
fof(t69_complex1,axiom,
$true ).
fof(t70_complex1,axiom,
$true ).
fof(t71_complex1,axiom,
$true ).
fof(t72_complex1,axiom,
k12_complex1(k7_complex1) = k10_complex1(k7_complex1) ).
fof(t73_complex1,axiom,
$true ).
fof(t74_complex1,axiom,
$true ).
fof(t75_complex1,axiom,
$true ).
fof(t76_complex1,axiom,
$true ).
fof(t77_complex1,axiom,
$true ).
fof(t78_complex1,axiom,
$true ).
fof(t79_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k4_complex1(A) = np__0
=> ( k3_complex1(A) = np__0
| ( k3_complex1(k12_complex1(A)) = k2_real_1(k3_complex1(A))
& k4_complex1(k12_complex1(A)) = np__0 ) ) ) ) ).
fof(t80_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(A) = np__0
=> ( k4_complex1(A) = np__0
| ( k3_complex1(k12_complex1(A)) = np__0
& k4_complex1(k12_complex1(A)) = k1_real_1(k2_real_1(k4_complex1(A))) ) ) ) ) ).
fof(d14_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k13_complex1(A,B) = k2_xcmplx_0(k6_real_1(k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B))),k3_real_1(k7_square_1(k3_complex1(B)),k7_square_1(k4_complex1(B)))),k3_xcmplx_0(k6_real_1(k5_real_1(k4_real_1(k3_complex1(B),k4_complex1(A)),k4_real_1(k3_complex1(A),k4_complex1(B))),k3_real_1(k7_square_1(k3_complex1(B)),k7_square_1(k4_complex1(B)))),k7_complex1)) ) ) ).
fof(t81_complex1,axiom,
$true ).
fof(t82_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k3_complex1(k13_complex1(A,B)) = k6_real_1(k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B))),k3_real_1(k7_square_1(k3_complex1(B)),k7_square_1(k4_complex1(B))))
& k4_complex1(k13_complex1(A,B)) = k6_real_1(k5_real_1(k4_real_1(k3_complex1(B),k4_complex1(A)),k4_real_1(k3_complex1(A),k4_complex1(B))),k3_real_1(k7_square_1(k3_complex1(B)),k7_square_1(k4_complex1(B)))) ) ) ) ).
fof(t83_complex1,axiom,
$true ).
fof(t84_complex1,axiom,
$true ).
fof(t85_complex1,axiom,
$true ).
fof(t86_complex1,axiom,
$true ).
fof(t87_complex1,axiom,
$true ).
fof(t88_complex1,axiom,
$true ).
fof(t89_complex1,axiom,
$true ).
fof(t90_complex1,axiom,
$true ).
fof(t91_complex1,axiom,
$true ).
fof(t92_complex1,axiom,
$true ).
fof(t93_complex1,axiom,
$true ).
fof(t94_complex1,axiom,
$true ).
fof(t95_complex1,axiom,
$true ).
fof(t96_complex1,axiom,
$true ).
fof(t97_complex1,axiom,
$true ).
fof(t98_complex1,axiom,
$true ).
fof(t99_complex1,axiom,
$true ).
fof(t100_complex1,axiom,
$true ).
fof(t101_complex1,axiom,
$true ).
fof(t102_complex1,axiom,
$true ).
fof(t103_complex1,axiom,
$true ).
fof(t104_complex1,axiom,
$true ).
fof(t105_complex1,axiom,
$true ).
fof(t106_complex1,axiom,
$true ).
fof(t107_complex1,axiom,
$true ).
fof(t108_complex1,axiom,
$true ).
fof(t109_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( ( k4_complex1(A) = np__0
& k4_complex1(B) = np__0 )
=> ( k3_complex1(B) = np__0
| ( k3_complex1(k13_complex1(A,B)) = k6_real_1(k3_complex1(A),k3_complex1(B))
& k4_complex1(k13_complex1(A,B)) = np__0 ) ) ) ) ) ).
fof(t110_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( ( k3_complex1(A) = np__0
& k3_complex1(B) = np__0 )
=> ( k4_complex1(B) = np__0
| ( k3_complex1(k13_complex1(A,B)) = k6_real_1(k4_complex1(A),k4_complex1(B))
& k4_complex1(k13_complex1(A,B)) = np__0 ) ) ) ) ) ).
fof(d15_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k14_complex1(A) = k6_xcmplx_0(k3_complex1(A),k3_xcmplx_0(k4_complex1(A),k7_complex1)) ) ).
fof(t111_complex1,axiom,
$true ).
fof(t112_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k3_complex1(k15_complex1(A)) = k3_complex1(A)
& k4_complex1(k15_complex1(A)) = k1_real_1(k4_complex1(A)) ) ) ).
fof(t113_complex1,axiom,
k15_complex1(np__0) = np__0 ).
fof(t114_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k15_complex1(A) = np__0
=> A = np__0 ) ) ).
fof(t115_complex1,axiom,
k15_complex1(k6_complex1) = k6_complex1 ).
fof(t116_complex1,axiom,
k15_complex1(k7_complex1) = k10_complex1(k7_complex1) ).
fof(t117_complex1,axiom,
$true ).
fof(t118_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k15_complex1(k2_xcmplx_0(A,B)) = k8_complex1(k15_complex1(A),k15_complex1(B)) ) ) ).
fof(t119_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k15_complex1(k10_complex1(A)) = k10_complex1(k15_complex1(A)) ) ).
fof(t120_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k15_complex1(k11_complex1(A,B)) = k11_complex1(k15_complex1(A),k15_complex1(B)) ) ) ).
fof(t121_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k15_complex1(k9_complex1(A,B)) = k9_complex1(k15_complex1(A),k15_complex1(B)) ) ) ).
fof(t122_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k15_complex1(k12_complex1(A)) = k12_complex1(k15_complex1(A)) ) ).
fof(t123_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k15_complex1(k13_complex1(A,B)) = k13_complex1(k15_complex1(A),k15_complex1(B)) ) ) ).
fof(t124_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k4_complex1(A) = np__0
=> k15_complex1(A) = A ) ) ).
fof(t125_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(A) = np__0
=> k15_complex1(A) = k10_complex1(A) ) ) ).
fof(t126_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(k9_complex1(A,k15_complex1(A))) = k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))
& k4_complex1(k9_complex1(A,k15_complex1(A))) = np__0 ) ) ).
fof(t127_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(k8_complex1(A,k15_complex1(A))) = k4_real_1(np__2,k3_complex1(A))
& k4_complex1(k8_complex1(A,k15_complex1(A))) = np__0 ) ) ).
fof(t128_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k3_complex1(k11_complex1(A,k15_complex1(A))) = np__0
& k4_complex1(k11_complex1(A,k15_complex1(A))) = k4_real_1(np__2,k4_complex1(A)) ) ) ).
fof(d16_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k16_complex1(A) = k9_square_1(k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A)))) ) ).
fof(t129_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k17_complex1(A) = A ) ) ).
fof(t130_complex1,axiom,
k17_complex1(np__0) = np__0 ).
fof(t131_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k17_complex1(A) = np__0
=> A = np__0 ) ) ).
fof(t132_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> r1_xreal_0(np__0,k17_complex1(A)) ) ).
fof(t133_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( ~ ( A != np__0
& r1_xreal_0(k17_complex1(A),np__0) )
& ~ ( ~ r1_xreal_0(k17_complex1(A),np__0)
& A = np__0 ) ) ) ).
fof(t134_complex1,axiom,
k17_complex1(k6_complex1) = np__1 ).
fof(t135_complex1,axiom,
k17_complex1(k7_complex1) = np__1 ).
fof(t136_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k4_complex1(A) = np__0
=> k17_complex1(A) = k17_complex1(k3_complex1(A)) ) ) ).
fof(t137_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( k3_complex1(A) = np__0
=> k17_complex1(A) = k17_complex1(k4_complex1(A)) ) ) ).
fof(t138_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ).
fof(t139_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k15_complex1(A)) = k17_complex1(A) ) ).
fof(t140_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> r1_xreal_0(k3_complex1(A),k17_complex1(A)) ) ).
fof(t141_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> r1_xreal_0(k4_complex1(A),k17_complex1(A)) ) ).
fof(t142_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> r1_xreal_0(k17_complex1(k2_xcmplx_0(A,B)),k3_real_1(k17_complex1(A),k17_complex1(B))) ) ) ).
fof(t143_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> r1_xreal_0(k17_complex1(k6_xcmplx_0(A,B)),k3_real_1(k17_complex1(A),k17_complex1(B))) ) ) ).
fof(t144_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> r1_xreal_0(k5_real_1(k17_complex1(A),k17_complex1(B)),k17_complex1(k2_xcmplx_0(A,B))) ) ) ).
fof(t145_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> r1_xreal_0(k5_real_1(k17_complex1(A),k17_complex1(B)),k17_complex1(k6_xcmplx_0(A,B))) ) ) ).
fof(t146_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k17_complex1(k6_xcmplx_0(A,B)) = k17_complex1(k6_xcmplx_0(B,A)) ) ) ).
fof(t147_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( k17_complex1(k6_xcmplx_0(A,B)) = np__0
<=> A = B ) ) ) ).
fof(t148_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( ~ ( A != B
& r1_xreal_0(k17_complex1(k6_xcmplx_0(A,B)),np__0) )
& ~ ( ~ r1_xreal_0(k17_complex1(k6_xcmplx_0(A,B)),np__0)
& A = B ) ) ) ) ).
fof(t149_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( v1_xcmplx_0(C)
=> r1_xreal_0(k17_complex1(k6_xcmplx_0(B,C)),k3_real_1(k17_complex1(k6_xcmplx_0(B,A)),k17_complex1(k6_xcmplx_0(A,C)))) ) ) ) ).
fof(t150_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> r1_xreal_0(k17_complex1(k5_real_1(k17_complex1(A),k17_complex1(B))),k17_complex1(k6_xcmplx_0(A,B))) ) ) ).
fof(t151_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k17_complex1(k3_xcmplx_0(A,B)) = k4_real_1(k17_complex1(A),k17_complex1(B)) ) ) ).
fof(t152_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k5_xcmplx_0(A)) = k2_real_1(k17_complex1(A)) ) ).
fof(t153_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k6_real_1(k17_complex1(A),k17_complex1(B)) = k17_complex1(k7_xcmplx_0(A,B)) ) ) ).
fof(t154_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k3_xcmplx_0(A,A)) = k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A))) ) ).
fof(t155_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k3_xcmplx_0(A,A)) = k17_complex1(k3_xcmplx_0(A,k15_complex1(A))) ) ).
fof(t156_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(A,np__0)
=> k17_complex1(A) = k4_xcmplx_0(A) ) ) ).
fof(t157_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k17_complex1(A) = A
| k17_complex1(A) = k4_xcmplx_0(A) ) ) ).
fof(t158_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> k8_square_1(k5_square_1(A)) = k17_complex1(A) ) ).
fof(t159_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k1_square_1(A,B) = k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(A,B),k17_complex1(k6_xcmplx_0(A,B))),np__2) ) ) ).
fof(t160_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_square_1(A,B) = k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(A,B),k17_complex1(k6_xcmplx_0(A,B))),np__2) ) ) ).
fof(t161_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> k7_square_1(k17_complex1(A)) = k5_square_1(A) ) ).
fof(t162_complex1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(k1_real_1(k17_complex1(A)),A)
& r1_xreal_0(A,k17_complex1(A)) ) ) ).
fof(t163_complex1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( k2_xcmplx_0(A,k3_xcmplx_0(B,k7_complex1)) = k2_xcmplx_0(C,k3_xcmplx_0(D,k7_complex1))
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(dt_k1_complex1,axiom,
$true ).
fof(dt_k2_complex1,axiom,
$true ).
fof(dt_k3_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k3_complex1(A),k1_numbers) ) ).
fof(redefinition_k3_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k3_complex1(A) = k1_complex1(A) ) ).
fof(dt_k4_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k4_complex1(A),k1_numbers) ) ).
fof(redefinition_k4_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k4_complex1(A) = k2_complex1(A) ) ).
fof(dt_k5_complex1,axiom,
m1_subset_1(k5_complex1,k2_numbers) ).
fof(dt_k6_complex1,axiom,
m1_subset_1(k6_complex1,k2_numbers) ).
fof(dt_k7_complex1,axiom,
m1_subset_1(k7_complex1,k2_numbers) ).
fof(redefinition_k7_complex1,axiom,
k7_complex1 = k1_xcmplx_0 ).
fof(dt_k8_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> m1_subset_1(k8_complex1(A,B),k2_numbers) ) ).
fof(commutativity_k8_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k8_complex1(A,B) = k8_complex1(B,A) ) ).
fof(redefinition_k8_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k8_complex1(A,B) = k2_xcmplx_0(A,B) ) ).
fof(dt_k9_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> m1_subset_1(k9_complex1(A,B),k2_numbers) ) ).
fof(commutativity_k9_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k9_complex1(A,B) = k9_complex1(B,A) ) ).
fof(redefinition_k9_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k9_complex1(A,B) = k3_xcmplx_0(A,B) ) ).
fof(dt_k10_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k10_complex1(A),k2_numbers) ) ).
fof(involutiveness_k10_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k10_complex1(k10_complex1(A)) = A ) ).
fof(redefinition_k10_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k10_complex1(A) = k4_xcmplx_0(A) ) ).
fof(dt_k11_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> m1_subset_1(k11_complex1(A,B),k2_numbers) ) ).
fof(redefinition_k11_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k11_complex1(A,B) = k6_xcmplx_0(A,B) ) ).
fof(dt_k12_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k12_complex1(A),k2_numbers) ) ).
fof(involutiveness_k12_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k12_complex1(k12_complex1(A)) = A ) ).
fof(redefinition_k12_complex1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k12_complex1(A) = k5_xcmplx_0(A) ) ).
fof(dt_k13_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> m1_subset_1(k13_complex1(A,B),k2_numbers) ) ).
fof(redefinition_k13_complex1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers) )
=> k13_complex1(A,B) = k7_xcmplx_0(A,B) ) ).
fof(dt_k14_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_xcmplx_0(k14_complex1(A)) ) ).
fof(involutiveness_k14_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k14_complex1(k14_complex1(A)) = A ) ).
fof(dt_k15_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k15_complex1(A),k2_numbers) ) ).
fof(involutiveness_k15_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k15_complex1(k15_complex1(A)) = A ) ).
fof(redefinition_k15_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k15_complex1(A) = k14_complex1(A) ) ).
fof(dt_k16_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> v1_xcmplx_0(k16_complex1(A)) ) ).
fof(projectivity_k16_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ).
fof(dt_k17_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k17_complex1(A),k1_numbers) ) ).
fof(projectivity_k17_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(k17_complex1(A)) = k17_complex1(A) ) ).
fof(redefinition_k17_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k17_complex1(A) = k16_complex1(A) ) ).
fof(dt_k18_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k18_complex1(A),k1_numbers) ) ).
fof(projectivity_k18_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ).
fof(redefinition_k18_complex1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k18_complex1(A) = k16_complex1(A) ) ).
%------------------------------------------------------------------------------