SET007 Axioms: SET007+585.ax


%------------------------------------------------------------------------------
% File     : SET007+585 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Ring of Integers, Euclidean Rings and Modulo 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   62 (   6 unit)
%            Number of atoms       :  495 (  65 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  501 (  68 ~  ;   7  |; 287  &)
%                                         (  13 <=>; 126 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   1 propositional; 0-3 arity)
%            Number of functors    :   47 (  14 constant; 0-6 arity)
%            Number of variables   :  112 (   0 singleton; 101 !;  11 ?)
%            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(fc1_int_3,axiom,
    ( ~ v3_struct_0(k1_int_3)
    & v3_vectsp_1(k1_int_3) )).

fof(fc2_int_3,axiom,
    ( ~ v3_struct_0(k1_int_3)
    & v2_group_1(k1_int_3)
    & v3_vectsp_1(k1_int_3)
    & v6_vectsp_1(k1_int_3)
    & v8_vectsp_1(k1_int_3) )).

fof(fc3_int_3,axiom,
    ( ~ v3_struct_0(k1_int_3)
    & v2_group_1(k1_int_3)
    & v4_group_1(k1_int_3)
    & v7_group_1(k1_int_3)
    & v3_rlvect_1(k1_int_3)
    & v4_rlvect_1(k1_int_3)
    & v5_rlvect_1(k1_int_3)
    & v6_rlvect_1(k1_int_3)
    & v3_vectsp_1(k1_int_3)
    & v4_vectsp_1(k1_int_3)
    & v5_vectsp_1(k1_int_3)
    & v6_vectsp_1(k1_int_3)
    & v7_vectsp_1(k1_int_3)
    & v8_vectsp_1(k1_int_3)
    & ~ v10_vectsp_1(k1_int_3)
    & v2_vectsp_2(k1_int_3) )).

fof(fc4_int_3,axiom,
    ( ~ v3_struct_0(k1_int_3)
    & v2_group_1(k1_int_3)
    & v4_group_1(k1_int_3)
    & v7_group_1(k1_int_3)
    & v3_rlvect_1(k1_int_3)
    & v4_rlvect_1(k1_int_3)
    & v5_rlvect_1(k1_int_3)
    & v6_rlvect_1(k1_int_3)
    & v3_vectsp_1(k1_int_3)
    & v4_vectsp_1(k1_int_3)
    & v5_vectsp_1(k1_int_3)
    & v6_vectsp_1(k1_int_3)
    & v7_vectsp_1(k1_int_3)
    & v8_vectsp_1(k1_int_3)
    & ~ v10_vectsp_1(k1_int_3)
    & v2_vectsp_2(k1_int_3)
    & v1_int_3(k1_int_3) )).

fof(rc1_int_3,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v3_vectsp_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v2_vectsp_2(A)
      & v1_int_3(A) ) )).

fof(rc2_int_3,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v3_vectsp_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v2_vectsp_2(A)
      & v1_int_3(A) ) )).

fof(cc1_int_3,axiom,(
    ! [A] :
      ( l3_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A)
          & v4_vectsp_1(A)
          & v6_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v2_vectsp_2(A)
          & v1_int_3(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A)
          & v4_vectsp_1(A)
          & v5_vectsp_1(A)
          & v6_vectsp_1(A)
          & v7_vectsp_1(A)
          & v8_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v2_vectsp_2(A)
          & v3_gcd_1(A) ) ) ) )).

fof(cc2_int_3,axiom,(
    ! [A] :
      ( l3_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v5_rlvect_1(A)
          & v8_vectsp_1(A)
          & v9_vectsp_1(A) )
       => ( ~ v3_struct_0(A)
          & v1_int_3(A) ) ) ) )).

fof(fc5_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ v3_struct_0(k9_int_3(A))
        & v3_vectsp_1(k9_int_3(A)) ) ) )).

fof(rc3_int_3,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v3_vectsp_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & v9_vectsp_1(A)
      & v10_vectsp_1(A)
      & v3_gcd_1(A)
      & v1_int_3(A) ) )).

fof(fc6_int_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m1_subset_1(A,k5_numbers) )
     => ( ~ v3_struct_0(k9_int_3(A))
        & v2_group_1(k9_int_3(A))
        & v4_group_1(k9_int_3(A))
        & v7_group_1(k9_int_3(A))
        & v3_rlvect_1(k9_int_3(A))
        & v4_rlvect_1(k9_int_3(A))
        & v5_rlvect_1(k9_int_3(A))
        & v6_rlvect_1(k9_int_3(A))
        & v3_vectsp_1(k9_int_3(A))
        & v4_vectsp_1(k9_int_3(A))
        & v5_vectsp_1(k9_int_3(A))
        & v6_vectsp_1(k9_int_3(A))
        & v7_vectsp_1(k9_int_3(A))
        & v8_vectsp_1(k9_int_3(A))
        & v9_vectsp_1(k9_int_3(A))
        & ~ v10_vectsp_1(k9_int_3(A))
        & v3_gcd_1(k9_int_3(A))
        & v1_int_3(k9_int_3(A)) ) ) )).

fof(d1_int_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers) )
     => ( A = k46_binop_2
      <=> ! [B] :
            ( m1_subset_1(B,k4_numbers)
           => ! [C] :
                ( m1_subset_1(C,k4_numbers)
               => k2_binop_1(k4_numbers,k4_numbers,k4_numbers,A,B,C) = k1_binop_1(k35_binop_2,B,C) ) ) ) ) )).

fof(d2_int_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k4_numbers,k4_numbers)
        & m2_relset_1(A,k4_numbers,k4_numbers) )
     => ( A = k43_binop_2
      <=> ! [B] :
            ( m1_subset_1(B,k4_numbers)
           => k8_funct_2(k4_numbers,k4_numbers,A,B) = k1_funct_1(k31_binop_2,B) ) ) ) )).

fof(d3_int_3,axiom,(
    k1_int_3 = g3_vectsp_1(k4_numbers,k44_binop_2,k46_binop_2,k1_funct_7(np__1,k4_numbers),k1_funct_7(np__0,k4_numbers)) )).

fof(d4_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_int_3))
         => ( r1_int_3(A,B)
          <=> ? [C] :
                ( v1_int_1(C)
                & ? [D] :
                    ( v1_int_1(D)
                    & C = A
                    & D = B
                    & r1_xreal_0(C,D) ) ) ) ) ) )).

fof(d5_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ( ( r1_int_3(k1_rlvect_1(k1_int_3),A)
         => k2_int_3(A) = A )
        & ( ~ r1_int_3(k1_rlvect_1(k1_int_3),A)
         => k2_int_3(A) = k5_rlvect_1(k1_int_3,A) ) ) ) )).

fof(d6_int_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k1_int_3),k5_numbers)
        & m2_relset_1(A,u1_struct_0(k1_int_3),k5_numbers) )
     => ( A = k3_int_3
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(k1_int_3))
           => k8_funct_2(u1_struct_0(k1_int_3),k5_numbers,A,B) = k1_funct_1(k2_euclid,B) ) ) ) )).

fof(t1_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => k8_funct_2(u1_struct_0(k1_int_3),k5_numbers,k3_int_3,A) = k2_int_3(A) ) )).

fof(t2_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_int_3))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_int_3))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k1_int_3))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k1_int_3))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k1_int_3))
                         => ( ( A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,C,B),E)
                              & r1_int_3(k1_rlvect_1(k1_int_3),E)
                              & A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,D,B),F)
                              & r1_int_3(k1_rlvect_1(k1_int_3),F) )
                           => ( B = k1_rlvect_1(k1_int_3)
                              | r1_int_3(k2_int_3(B),E)
                              | r1_int_3(k2_int_3(B),F)
                              | ( C = D
                                & E = F ) ) ) ) ) ) ) ) ) )).

fof(d7_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_int_3))
         => ( B != k1_rlvect_1(k1_int_3)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k1_int_3))
               => ( C = k4_int_3(A,B)
                <=> ? [D] :
                      ( m1_subset_1(D,u1_struct_0(k1_int_3))
                      & A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,C,B),D)
                      & r1_int_3(k1_rlvect_1(k1_int_3),D)
                      & ~ r1_int_3(k2_int_3(B),D) ) ) ) ) ) ) )).

fof(d8_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_int_3))
         => ( B != k1_rlvect_1(k1_int_3)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k1_int_3))
               => ( C = k5_int_3(A,B)
                <=> ? [D] :
                      ( m1_subset_1(D,u1_struct_0(k1_int_3))
                      & A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,D,B),C)
                      & r1_int_3(k1_rlvect_1(k1_int_3),C)
                      & ~ r1_int_3(k2_int_3(B),C) ) ) ) ) ) ) )).

fof(t3_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_int_3))
         => ( B != k1_rlvect_1(k1_int_3)
           => A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,k4_int_3(A,B),B),k5_int_3(A,B)) ) ) ) )).

fof(d9_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v1_int_3(A)
      <=> ? [B] :
            ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k5_numbers)
            & m2_relset_1(B,u1_struct_0(A),k5_numbers)
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ~ ( D != k1_rlvect_1(A)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ~ ( C = k2_rlvect_1(A,k1_group_1(A,E,D),F)
                                    & ~ ( F != k1_rlvect_1(A)
                                        & r1_xreal_0(k8_funct_2(u1_struct_0(A),k5_numbers,B,D),k8_funct_2(u1_struct_0(A),k5_numbers,B,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(d10_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_int_3(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k5_numbers)
            & m2_relset_1(B,u1_struct_0(A),k5_numbers) )
         => ( m1_int_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ~ ( D != k1_rlvect_1(A)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ~ ( C = k2_rlvect_1(A,k1_group_1(A,E,D),F)
                                    & ~ ( F != k1_rlvect_1(A)
                                        & r1_xreal_0(k8_funct_2(u1_struct_0(A),k5_numbers,B,D),k8_funct_2(u1_struct_0(A),k5_numbers,B,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(A)
        & v1_int_3(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(A)
        & v3_gcd_1(A)
        & l3_vectsp_1(A) ) ) )).

fof(t5_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v5_rlvect_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & l3_vectsp_1(A) )
     => v1_int_3(A) ) )).

fof(t6_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v5_rlvect_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k5_numbers)
            & m2_relset_1(B,u1_struct_0(A),k5_numbers) )
         => m1_int_3(B,A) ) ) )).

fof(t7_int_3,axiom,(
    $true )).

fof(t8_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( A != np__0
                 => k5_int_1(k2_xcmplx_0(B,k3_xcmplx_0(A,C)),A) = k2_xcmplx_0(k5_int_1(B,A),C) )
                & k6_int_1(k2_xcmplx_0(B,k3_xcmplx_0(A,C)),A) = k6_int_1(B,A) ) ) ) ) )).

fof(t9_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( v1_int_1(B)
           => ( r1_xreal_0(np__0,k6_int_1(B,A))
              & ~ r1_xreal_0(A,k6_int_1(B,A)) ) ) ) ) )).

fof(t10_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(np__0,B)
             => ( r1_xreal_0(A,B)
                | k6_int_1(B,A) = B ) )
            & ( r1_xreal_0(k4_xcmplx_0(A),B)
             => ( r1_xreal_0(np__0,B)
                | k6_int_1(B,A) = k2_xcmplx_0(A,B) ) ) ) ) ) )).

fof(t11_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( v1_int_1(B)
           => ( k6_int_1(B,A) = np__0
            <=> r2_int_1(A,B) ) ) ) ) )).

fof(t12_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( k6_int_1(B,A) = k6_int_1(C,A)
                 => ( A = np__0
                    | r1_int_1(B,C,A) ) )
                & ( r1_int_1(B,C,A)
                 => k6_int_1(B,A) = k6_int_1(C,A) ) ) ) ) ) )).

fof(t13_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_int_1(B)
         => k6_int_1(k6_int_1(B,A),A) = k6_int_1(B,A) ) ) )).

fof(t14_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => k6_int_1(k2_xcmplx_0(B,C),A) = k6_int_1(k2_xcmplx_0(k6_int_1(B,A),k6_int_1(C,A)),A) ) ) ) )).

fof(t15_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => k6_int_1(k3_xcmplx_0(B,C),A) = k6_int_1(k3_xcmplx_0(k6_int_1(B,A),k6_int_1(C,A)),A) ) ) ) )).

fof(t16_int_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ? [C] :
              ( v1_int_1(C)
              & ? [D] :
                  ( v1_int_1(D)
                  & k3_int_2(A,B) = k2_xcmplx_0(k3_xcmplx_0(C,A),k3_xcmplx_0(D,B)) ) ) ) ) )).

fof(d11_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
              & m2_relset_1(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) )
           => ( B = k7_int_3(A)
            <=> ! [C] :
                  ( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
                 => ! [D] :
                      ( m2_subset_1(D,k5_numbers,k1_gr_cy_1(A))
                     => k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),B,C,D) = k4_nat_1(k2_nat_1(C,D),A) ) ) ) ) ) ) )).

fof(d12_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k1_gr_cy_1(A),k1_gr_cy_1(A))
              & m2_relset_1(B,k1_gr_cy_1(A),k1_gr_cy_1(A)) )
           => ( B = k8_int_3(A)
            <=> ! [C] :
                  ( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
                 => k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),B,C) = k6_int_1(k6_xcmplx_0(A,C),A) ) ) ) ) ) )).

fof(t17_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
               => ( ( ~ r1_xreal_0(A,k1_nat_1(B,C))
                   => k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k1_nat_1(B,C) )
                  & ~ ( k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k1_nat_1(B,C)
                      & r1_xreal_0(A,k1_nat_1(B,C)) )
                  & ( r1_xreal_0(A,k1_nat_1(B,C))
                   => k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k6_xcmplx_0(k1_nat_1(B,C),A) )
                  & ( k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k6_xcmplx_0(k1_nat_1(B,C),A)
                   => r1_xreal_0(A,k1_nat_1(B,C)) ) ) ) ) ) ) )).

fof(t18_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
           => ! [C] :
                ( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
               => ! [D] :
                    ( v4_ordinal2(D)
                   => ( ( r1_xreal_0(k3_xcmplx_0(D,A),k2_nat_1(B,C))
                        & ~ r1_xreal_0(k3_xcmplx_0(k2_xcmplx_0(D,np__1),A),k2_nat_1(B,C)) )
                    <=> k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k7_int_3(A),B,C) = k6_xcmplx_0(k2_nat_1(B,C),k3_xcmplx_0(D,A)) ) ) ) ) ) ) )).

fof(t19_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
           => ( ( B = np__0
               => k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = np__0 )
              & ( k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = np__0
               => B = np__0 )
              & ( B != np__0
               => k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = k6_xcmplx_0(A,B) )
              & ~ ( k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = k6_xcmplx_0(A,B)
                  & B = np__0 ) ) ) ) ) )).

fof(d13_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k9_int_3(A) = g3_vectsp_1(k1_gr_cy_1(A),k4_gr_cy_1(A),k7_int_3(A),k1_funct_7(np__1,k1_gr_cy_1(A)),k1_funct_7(np__0,k1_gr_cy_1(A))) ) )).

fof(t20_int_3,axiom,
    ( v10_vectsp_1(k9_int_3(np__1))
    & ~ v3_struct_0(k9_int_3(np__1))
    & v4_group_1(k9_int_3(np__1))
    & v3_rlvect_1(k9_int_3(np__1))
    & v4_rlvect_1(k9_int_3(np__1))
    & v5_rlvect_1(k9_int_3(np__1))
    & v6_rlvect_1(k9_int_3(np__1))
    & v6_vectsp_1(k9_int_3(np__1))
    & v7_vectsp_1(k9_int_3(np__1))
    & v8_vectsp_1(k9_int_3(np__1))
    & l3_vectsp_1(k9_int_3(np__1))
    & v9_vectsp_1(k9_int_3(np__1))
    & v2_group_1(k9_int_3(np__1))
    & v7_vectsp_1(k9_int_3(np__1))
    & v7_group_1(k9_int_3(np__1)) )).

fof(t21_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => ( ~ v10_vectsp_1(k9_int_3(A))
          & ~ v3_struct_0(k9_int_3(A))
          & v2_group_1(k9_int_3(A))
          & v4_group_1(k9_int_3(A))
          & v7_group_1(k9_int_3(A))
          & v3_rlvect_1(k9_int_3(A))
          & v4_rlvect_1(k9_int_3(A))
          & v5_rlvect_1(k9_int_3(A))
          & v6_rlvect_1(k9_int_3(A))
          & v6_vectsp_1(k9_int_3(A))
          & v7_vectsp_1(k9_int_3(A))
          & v8_vectsp_1(k9_int_3(A))
          & l3_vectsp_1(k9_int_3(A)) ) ) ) )).

fof(t22_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => ( ( ~ v3_struct_0(k9_int_3(A))
            & v4_group_1(k9_int_3(A))
            & v7_group_1(k9_int_3(A))
            & v3_rlvect_1(k9_int_3(A))
            & v4_rlvect_1(k9_int_3(A))
            & v5_rlvect_1(k9_int_3(A))
            & v6_rlvect_1(k9_int_3(A))
            & v7_vectsp_1(k9_int_3(A))
            & v8_vectsp_1(k9_int_3(A))
            & v9_vectsp_1(k9_int_3(A))
            & ~ v10_vectsp_1(k9_int_3(A))
            & l3_vectsp_1(k9_int_3(A)) )
        <=> ( v1_int_2(A)
            & m2_subset_1(A,k1_numbers,k5_numbers) ) ) ) ) )).

fof(t23_int_3,axiom,(
    k2_group_1(k1_int_3) = np__1 )).

fof(t24_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => k2_group_1(k9_int_3(A)) = np__1 ) ) )).

fof(dt_m1_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_int_3(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_int_3(B,A)
         => ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k5_numbers)
            & m2_relset_1(B,u1_struct_0(A),k5_numbers) ) ) ) )).

fof(existence_m1_int_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_int_3(A)
        & l3_vectsp_1(A) )
     => ? [B] : m1_int_3(B,A) ) )).

fof(reflexivity_r1_int_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,u1_struct_0(k1_int_3))
        & m1_subset_1(B,u1_struct_0(k1_int_3)) )
     => r1_int_3(A,A) ) )).

fof(connectedness_r1_int_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,u1_struct_0(k1_int_3))
        & m1_subset_1(B,u1_struct_0(k1_int_3)) )
     => ( r1_int_3(A,B)
        | r1_int_3(B,A) ) ) )).

fof(dt_k1_int_3,axiom,(
    l3_vectsp_1(k1_int_3) )).

fof(dt_k2_int_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_int_3))
     => m1_subset_1(k2_int_3(A),u1_struct_0(k1_int_3)) ) )).

fof(dt_k3_int_3,axiom,
    ( v1_funct_1(k3_int_3)
    & v1_funct_2(k3_int_3,u1_struct_0(k1_int_3),k5_numbers)
    & m2_relset_1(k3_int_3,u1_struct_0(k1_int_3),k5_numbers) )).

fof(dt_k4_int_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,u1_struct_0(k1_int_3))
        & m1_subset_1(B,u1_struct_0(k1_int_3)) )
     => m1_subset_1(k4_int_3(A,B),u1_struct_0(k1_int_3)) ) )).

fof(dt_k5_int_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,u1_struct_0(k1_int_3))
        & m1_subset_1(B,u1_struct_0(k1_int_3)) )
     => m1_subset_1(k5_int_3(A,B),u1_struct_0(k1_int_3)) ) )).

fof(dt_k6_int_3,axiom,(
    m1_int_3(k6_int_3,k1_int_3) )).

fof(redefinition_k6_int_3,axiom,(
    k6_int_3 = k3_int_3 )).

fof(dt_k7_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_funct_1(k7_int_3(A))
        & v1_funct_2(k7_int_3(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
        & m2_relset_1(k7_int_3(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) ) ) )).

fof(dt_k8_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_funct_1(k8_int_3(A))
        & v1_funct_2(k8_int_3(A),k1_gr_cy_1(A),k1_gr_cy_1(A))
        & m2_relset_1(k8_int_3(A),k1_gr_cy_1(A),k1_gr_cy_1(A)) ) ) )).

fof(dt_k9_int_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => l3_vectsp_1(k9_int_3(A)) ) )).
%------------------------------------------------------------------------------