SET007 Axioms: SET007+48.ax


%------------------------------------------------------------------------------
% File     : SET007+48 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Integers
% 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    : int_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  121 (  29 unit)
%            Number of atoms       :  430 (  53 equality)
%            Maximal formula depth :   13 (   5 average)
%            Number of connectives :  351 (  42 ~  ;   7  |;  97  &)
%                                         (  18 <=>; 187 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (  10 constant; 0-2 arity)
%            Number of variables   :  173 (   0 singleton; 164 !;   9 ?)
%            Maximal term depth    :    4 (   1 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(rc1_int_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_int_1(A) ) )).

fof(rc2_int_1,axiom,(
    ? [A] : v1_int_1(A) )).

fof(cc1_int_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_numbers)
     => v1_int_1(A) ) )).

fof(cc2_int_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_int_1(A) ) ) )).

fof(cc3_int_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v1_int_1(A) ) )).

fof(cc4_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) )).

fof(fc1_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc2_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc3_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & v1_int_1(k4_xcmplx_0(A)) ) ) )).

fof(fc4_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc5_int_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & ~ v2_xreal_0(k4_xcmplx_0(A))
        & v1_int_1(k4_xcmplx_0(A)) ) ) )).

fof(fc6_int_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(B,A))
        & v1_xreal_0(k2_xcmplx_0(B,A))
        & v1_int_1(k2_xcmplx_0(B,A)) ) ) )).

fof(fc7_int_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(B,A))
        & v1_xreal_0(k3_xcmplx_0(B,A))
        & v1_int_1(k3_xcmplx_0(B,A)) ) ) )).

fof(fc8_int_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & v1_int_1(k6_xcmplx_0(B,A)) ) ) )).

fof(fc9_int_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc10_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k3_int_1(A))
        & v1_xreal_0(k3_int_1(A)) ) ) )).

fof(d1_int_1,axiom,(
    ! [A] :
      ( A = k4_numbers
    <=> ! [B] :
          ( r2_hidden(B,A)
        <=> ~ ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( B != C
                  & B != k1_real_1(C) ) ) ) ) )).

fof(d2_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
    <=> r2_hidden(A,k4_numbers) ) )).

fof(t1_int_1,axiom,(
    $true )).

fof(t2_int_1,axiom,(
    $true )).

fof(t3_int_1,axiom,(
    $true )).

fof(t4_int_1,axiom,(
    $true )).

fof(t5_int_1,axiom,(
    $true )).

fof(t6_int_1,axiom,(
    $true )).

fof(t7_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( A = B
              | A = k4_xcmplx_0(B) )
           => v1_int_1(A) ) ) ) )).

fof(t8_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( v1_int_1(A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( A != B
                & A != k1_real_1(B) ) ) ) ) )).

fof(t9_int_1,axiom,(
    $true )).

fof(t10_int_1,axiom,(
    $true )).

fof(t11_int_1,axiom,(
    $true )).

fof(t12_int_1,axiom,(
    $true )).

fof(t13_int_1,axiom,(
    $true )).

fof(t14_int_1,axiom,(
    $true )).

fof(t15_int_1,axiom,(
    $true )).

fof(t16_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( r1_xreal_0(np__0,A)
       => r2_hidden(A,k5_numbers) ) ) )).

fof(t17_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_int_1(A)
       => ( v1_int_1(k2_xcmplx_0(A,np__1))
          & v1_int_1(k6_xcmplx_0(A,np__1)) ) ) ) )).

fof(t18_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(A,B)
           => r2_hidden(k6_xcmplx_0(B,A),k5_numbers) ) ) ) )).

fof(t19_int_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( k2_xcmplx_0(B,A) = C
               => r1_xreal_0(B,C) ) ) ) ) )).

fof(t20_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ~ r1_xreal_0(B,A)
           => r1_xreal_0(k2_xcmplx_0(A,np__1),B) ) ) ) )).

fof(t21_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( ~ r1_xreal_0(np__0,A)
       => r1_xreal_0(A,k1_real_1(np__1)) ) ) )).

fof(t22_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( k3_xcmplx_0(A,B) = np__1
          <=> ( ( A = np__1
                & B = np__1 )
              | ( A = k1_real_1(np__1)
                & B = k1_real_1(np__1) ) ) ) ) ) )).

fof(t23_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( k3_xcmplx_0(A,B) = k1_real_1(np__1)
          <=> ( ( A = k1_real_1(np__1)
                & B = np__1 )
              | ( A = np__1
                & B = k1_real_1(np__1) ) ) ) ) ) )).

fof(d3_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(A,B,C)
              <=> ? [D] :
                    ( v1_int_1(D)
                    & k3_xcmplx_0(C,D) = k6_xcmplx_0(A,B) ) ) ) ) ) )).

fof(t24_int_1,axiom,(
    $true )).

fof(t25_int_1,axiom,(
    $true )).

fof(t26_int_1,axiom,(
    $true )).

fof(t27_int_1,axiom,(
    $true )).

fof(t28_int_1,axiom,(
    $true )).

fof(t29_int_1,axiom,(
    $true )).

fof(t30_int_1,axiom,(
    $true )).

fof(t31_int_1,axiom,(
    $true )).

fof(t32_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r1_int_1(A,A,B) ) ) )).

fof(t33_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( r1_int_1(A,np__0,A)
        & r1_int_1(np__0,A,A) ) ) )).

fof(t34_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r1_int_1(A,B,np__1) ) ) )).

fof(t35_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(A,B,C)
               => r1_int_1(B,A,C) ) ) ) ) )).

fof(t36_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ( ( r1_int_1(A,B,C)
                      & r1_int_1(B,D,C) )
                   => r1_int_1(A,D,C) ) ) ) ) ) )).

fof(t37_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( ( r1_int_1(A,B,C)
                          & r1_int_1(D,E,C) )
                       => r1_int_1(k2_xcmplx_0(A,D),k2_xcmplx_0(B,E),C) ) ) ) ) ) ) )).

fof(t38_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( ( r1_int_1(A,B,C)
                          & r1_int_1(D,E,C) )
                       => r1_int_1(k6_xcmplx_0(A,D),k6_xcmplx_0(B,E),C) ) ) ) ) ) ) )).

fof(t39_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( ( r1_int_1(A,B,C)
                          & r1_int_1(D,E,C) )
                       => r1_int_1(k3_xcmplx_0(A,D),k3_xcmplx_0(B,E),C) ) ) ) ) ) ) )).

fof(t40_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ( r1_int_1(k2_xcmplx_0(A,B),C,D)
                  <=> r1_int_1(A,k6_xcmplx_0(C,B),D) ) ) ) ) ) )).

fof(t41_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( ( k3_xcmplx_0(A,B) = C
                          & r1_int_1(D,E,C) )
                       => r1_int_1(D,E,A) ) ) ) ) ) ) )).

fof(t42_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(A,B,C)
              <=> r1_int_1(k2_xcmplx_0(A,C),B,C) ) ) ) ) )).

fof(t43_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(A,B,C)
              <=> r1_int_1(k6_xcmplx_0(A,C),B,C) ) ) ) ) )).

fof(t44_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_xreal_0(B,A)
                  & r1_xreal_0(C,A) )
               => ( r1_xreal_0(B,k6_xcmplx_0(A,np__1))
                  | r1_xreal_0(C,k6_xcmplx_0(A,np__1))
                  | B = C ) ) ) ) ) )).

fof(t45_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(A,C) )
               => ( r1_xreal_0(k2_xcmplx_0(A,np__1),B)
                  | r1_xreal_0(k2_xcmplx_0(A,np__1),C)
                  | B = C ) ) ) ) ) )).

fof(d4_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( B = k1_int_1(A)
          <=> ( r1_xreal_0(B,A)
              & ~ r1_xreal_0(B,k6_xcmplx_0(A,np__1)) ) ) ) ) )).

fof(t46_int_1,axiom,(
    $true )).

fof(t47_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k1_int_1(A) = A
      <=> v1_int_1(A) ) ) )).

fof(t48_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ ( ~ r1_xreal_0(A,k1_int_1(A))
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(A,k1_int_1(A)) ) ) ) )).

fof(t49_int_1,axiom,(
    $true )).

fof(t50_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,k6_xcmplx_0(k1_int_1(A),np__1))
        & ~ r1_xreal_0(k2_xcmplx_0(A,np__1),k1_int_1(A)) ) ) )).

fof(t51_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => k2_xcmplx_0(k1_int_1(A),B) = k1_int_1(k2_xcmplx_0(A,B)) ) ) )).

fof(t52_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ r1_xreal_0(k2_xcmplx_0(k1_int_1(A),np__1),A) ) )).

fof(d5_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( B = k2_int_1(A)
          <=> ( r1_xreal_0(A,B)
              & ~ r1_xreal_0(k2_xcmplx_0(A,np__1),B) ) ) ) ) )).

fof(t53_int_1,axiom,(
    $true )).

fof(t54_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k2_int_1(A) = A
      <=> v1_int_1(A) ) ) )).

fof(t55_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ ( ~ r1_xreal_0(k2_int_1(A),A)
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(k2_int_1(A),A) ) ) ) )).

fof(t56_int_1,axiom,(
    $true )).

fof(t57_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(k2_int_1(A),k6_xcmplx_0(A,np__1))
        & ~ r1_xreal_0(k2_xcmplx_0(k2_int_1(A),np__1),A) ) ) )).

fof(t58_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => k2_xcmplx_0(k2_int_1(A),B) = k2_int_1(k2_xcmplx_0(A,B)) ) ) )).

fof(t59_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k1_int_1(A) = k2_int_1(A)
      <=> v1_int_1(A) ) ) )).

fof(t60_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ ( ~ r1_xreal_0(k2_int_1(A),k1_int_1(A))
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(k2_int_1(A),k1_int_1(A)) ) ) ) )).

fof(t61_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => r1_xreal_0(k1_int_1(A),k2_int_1(A)) ) )).

fof(t62_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_int_1(k2_int_1(A)) = k2_int_1(A) ) )).

fof(t63_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_int_1(k1_int_1(A)) = k1_int_1(A) ) )).

fof(t64_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_int_1(k2_int_1(A)) = k2_int_1(A) ) )).

fof(t65_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_int_1(k1_int_1(A)) = k1_int_1(A) ) )).

fof(t66_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k1_int_1(A) = k2_int_1(A)
      <=> k2_xcmplx_0(k1_int_1(A),np__1) != k2_int_1(A) ) ) )).

fof(d6_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_int_1(A) = k6_xcmplx_0(A,k1_int_1(A)) ) )).

fof(t67_int_1,axiom,(
    $true )).

fof(t68_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => A = k2_xcmplx_0(k1_int_1(A),k4_int_1(A)) ) )).

fof(t69_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(np__1,k4_int_1(A))
        & r1_xreal_0(np__0,k4_int_1(A)) ) ) )).

fof(t70_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_int_1(k4_int_1(A)) = np__0 ) )).

fof(t71_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k4_int_1(A) = np__0
      <=> v1_int_1(A) ) ) )).

fof(t72_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ ( ~ r1_xreal_0(k4_int_1(A),np__0)
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(k4_int_1(A),np__0) ) ) ) )).

fof(d7_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => k5_int_1(A,B) = k1_int_1(k7_xcmplx_0(A,B)) ) ) )).

fof(d8_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( B != np__0
             => k6_int_1(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k5_int_1(A,B),B)) )
            & ( B = np__0
             => k6_int_1(A,B) = np__0 ) ) ) ) )).

fof(d9_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r2_int_1(A,B)
          <=> ? [C] :
                ( v1_int_1(C)
                & B = k3_xcmplx_0(A,C) ) ) ) ) )).

fof(t73_int_1,axiom,(
    $true )).

fof(t74_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A != np__0
       => k1_int_1(k7_xcmplx_0(A,A)) = np__1 ) ) )).

fof(t75_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k5_int_1(A,np__0) = np__0 ) )).

fof(t76_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( A != np__0
       => k5_int_1(A,A) = np__1 ) ) )).

fof(t77_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k6_int_1(A,A) = np__0 ) )).

fof(t78_int_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ r1_xreal_0(B,A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C = k6_xcmplx_0(B,A)
                      & r1_xreal_0(np__1,C) ) ) ) ) ) )).

fof(t79_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ~ r1_xreal_0(B,A)
           => r1_xreal_0(A,k6_xcmplx_0(B,np__1)) ) ) ) )).

fof(t80_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( r1_xreal_0(np__0,A)
       => ( r1_xreal_0(np__0,k2_int_1(A))
          & r1_xreal_0(np__0,k1_int_1(A))
          & m2_subset_1(k2_int_1(A),k1_numbers,k5_numbers)
          & m2_subset_1(k1_int_1(A),k1_numbers,k5_numbers) ) ) ) )).

fof(t81_int_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => r1_xreal_0(A,k1_int_1(B)) ) ) ) )).

fof(t82_int_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => r1_xreal_0(np__0,k5_int_1(A,B)) ) ) )).

fof(s1_int_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k4_numbers))
      & ! [B] :
          ( v1_int_1(B)
         => ( r2_hidden(B,A)
          <=> p1_s1_int_1(B) ) ) ) )).

fof(s2_int_1,axiom,
    ( ( p1_s2_int_1(f1_s2_int_1)
      & ! [A] :
          ( v1_int_1(A)
         => ( ( r1_xreal_0(f1_s2_int_1,A)
              & p1_s2_int_1(A) )
           => p1_s2_int_1(k2_xcmplx_0(A,np__1)) ) ) )
   => ! [A] :
        ( v1_int_1(A)
       => ( r1_xreal_0(f1_s2_int_1,A)
         => p1_s2_int_1(A) ) ) )).

fof(s3_int_1,axiom,
    ( ( p1_s3_int_1(f1_s3_int_1)
      & ! [A] :
          ( v1_int_1(A)
         => ( ( r1_xreal_0(A,f1_s3_int_1)
              & p1_s3_int_1(A) )
           => p1_s3_int_1(k6_xcmplx_0(A,np__1)) ) ) )
   => ! [A] :
        ( v1_int_1(A)
       => ( r1_xreal_0(A,f1_s3_int_1)
         => p1_s3_int_1(A) ) ) )).

fof(s4_int_1,axiom,
    ( ( p1_s4_int_1(f1_s4_int_1)
      & ! [A] :
          ( v1_int_1(A)
         => ( p1_s4_int_1(A)
           => ( p1_s4_int_1(k6_xcmplx_0(A,np__1))
              & p1_s4_int_1(k2_xcmplx_0(A,np__1)) ) ) ) )
   => ! [A] :
        ( v1_int_1(A)
       => p1_s4_int_1(A) ) )).

fof(s5_int_1,axiom,
    ( ( ! [A] :
          ( v1_int_1(A)
         => ( p1_s5_int_1(A)
           => r1_xreal_0(f1_s5_int_1,A) ) )
      & ? [A] :
          ( v1_int_1(A)
          & p1_s5_int_1(A) ) )
   => ? [A] :
        ( v1_int_1(A)
        & p1_s5_int_1(A)
        & ! [B] :
            ( v1_int_1(B)
           => ( p1_s5_int_1(B)
             => r1_xreal_0(A,B) ) ) ) )).

fof(s6_int_1,axiom,
    ( ( ! [A] :
          ( v1_int_1(A)
         => ( p1_s6_int_1(A)
           => r1_xreal_0(A,f1_s6_int_1) ) )
      & ? [A] :
          ( v1_int_1(A)
          & p1_s6_int_1(A) ) )
   => ? [A] :
        ( v1_int_1(A)
        & p1_s6_int_1(A)
        & ! [B] :
            ( v1_int_1(B)
           => ( p1_s6_int_1(B)
             => r1_xreal_0(B,A) ) ) ) )).

fof(reflexivity_r2_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => r2_int_1(A,A) ) )).

fof(dt_k1_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v1_int_1(k1_int_1(A)) ) )).

fof(dt_k2_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v1_int_1(k2_int_1(A)) ) )).

fof(dt_k3_int_1,axiom,(
    $true )).

fof(dt_k4_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => m1_subset_1(k4_int_1(A),k1_numbers) ) )).

fof(redefinition_k4_int_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k4_int_1(A) = k3_int_1(A) ) )).

fof(dt_k5_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k5_int_1(A,B)) ) )).

fof(dt_k6_int_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k6_int_1(A,B)) ) )).
%------------------------------------------------------------------------------