SET007 Axioms: SET007+52.ax


%------------------------------------------------------------------------------
% File     : SET007+52 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  478 ( 165 equality)
%            Maximal formula depth :   12 (   3 average)
%            Number of connectives :  265 (  10 ~  ;   5  |;  69  &)
%                                         (   8 <=>; 173 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   45 (   9 constant; 0-5 arity)
%            Number of variables   :  169 (   0 singleton; 165 !;   4 ?)
%            Maximal term depth    :    7 (   2 average)
% 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) ) )).
%------------------------------------------------------------------------------