SET007 Axioms: SET007+47.ax


%------------------------------------------------------------------------------
% File     : SET007+47 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Fundamental Properties of Natural 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    : nat_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  116 (  29 unit)
%            Number of atoms       :  473 (  80 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  420 (  63 ~  ;   6  |; 126  &)
%                                         (   9 <=>; 216 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   1 propositional; 0-3 arity)
%            Number of functors    :   22 (   9 constant; 0-2 arity)
%            Number of variables   :  203 (   0 singleton; 190 !;  13 ?)
%            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_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( v4_ordinal2(k2_xcmplx_0(A,B))
        & v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc2_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( v4_ordinal2(k3_xcmplx_0(A,B))
        & v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) )).

fof(rc1_nat_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) )).

fof(fc3_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & ~ v1_xboole_0(B)
        & v4_ordinal2(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
        & v4_ordinal2(k2_xcmplx_0(A,B))
        & v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc4_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & ~ v1_xboole_0(B)
        & v4_ordinal2(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
        & v4_ordinal2(k2_xcmplx_0(B,A))
        & v1_xcmplx_0(k2_xcmplx_0(B,A))
        & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) )).

fof(cc1_nat_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) ) ) )).

fof(rc2_nat_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v3_ordinal1(A) ) )).

fof(rc3_nat_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) )).

fof(cc2_nat_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) ) ) )).

fof(cc3_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) )).

fof(t1_nat_1,axiom,(
    $true )).

fof(t2_nat_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( r2_hidden(np__0,A)
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ( r2_hidden(B,A)
               => r2_hidden(k3_real_1(B,np__1),A) ) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r2_hidden(B,A) ) ) ) )).

fof(t3_nat_1,axiom,(
    $true )).

fof(t4_nat_1,axiom,(
    $true )).

fof(t5_nat_1,axiom,(
    $true )).

fof(t6_nat_1,axiom,(
    $true )).

fof(t7_nat_1,axiom,(
    $true )).

fof(t8_nat_1,axiom,(
    $true )).

fof(t9_nat_1,axiom,(
    $true )).

fof(t10_nat_1,axiom,(
    $true )).

fof(t11_nat_1,axiom,(
    $true )).

fof(t12_nat_1,axiom,(
    $true )).

fof(t13_nat_1,axiom,(
    $true )).

fof(t14_nat_1,axiom,(
    $true )).

fof(t15_nat_1,axiom,(
    $true )).

fof(t16_nat_1,axiom,(
    $true )).

fof(t17_nat_1,axiom,(
    $true )).

fof(t18_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => r1_xreal_0(np__0,A) ) )).

fof(t19_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( np__0 != A
          & r1_xreal_0(A,np__0) ) ) )).

fof(t20_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t21_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ~ r1_xreal_0(k2_xcmplx_0(A,np__1),np__0) ) )).

fof(t22_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( A != np__0
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => A != k1_nat_1(B,np__1) ) ) ) )).

fof(t23_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( k2_xcmplx_0(A,B) = np__0
           => ( A = np__0
              & B = np__0 ) ) ) ) )).

fof(t24_nat_1,axiom,(
    $true )).

fof(t25_nat_1,axiom,(
    $true )).

fof(t26_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,np__1))
              & ~ r1_xreal_0(A,B)
              & A != k2_xcmplx_0(B,np__1) ) ) ) )).

fof(t27_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,B)
              & r1_xreal_0(B,k2_xcmplx_0(A,np__1))
              & A != B
              & B != k2_xcmplx_0(A,np__1) ) ) ) )).

fof(t28_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k2_xcmplx_0(A,C) ) ) ) ) )).

fof(t29_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) )).

fof(t30_nat_1,axiom,(
    $true )).

fof(t31_nat_1,axiom,(
    $true )).

fof(t32_nat_1,axiom,(
    $true )).

fof(t33_nat_1,axiom,(
    $true )).

fof(t34_nat_1,axiom,(
    $true )).

fof(t35_nat_1,axiom,(
    $true )).

fof(t36_nat_1,axiom,(
    $true )).

fof(t37_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) )).

fof(t38_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(B,np__1),A)
          <=> r1_xreal_0(A,B) ) ) ) )).

fof(t39_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(np__1,A)
       => A = np__0 ) ) )).

fof(t40_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( k3_xcmplx_0(A,B) = np__1
           => ( A = np__1
              & B = np__1 ) ) ) ) )).

fof(t41_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( A != np__0
              & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) )).

fof(t42_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & B = k1_nat_1(k2_nat_1(A,C),D)
                    & ~ r1_xreal_0(A,D) ) ) ) ) ) )).

fof(t43_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ! [E] :
                      ( v4_ordinal2(E)
                     => ! [F] :
                          ( v4_ordinal2(F)
                         => ( ( A = k2_xcmplx_0(k3_xcmplx_0(B,C),E)
                              & A = k2_xcmplx_0(k3_xcmplx_0(B,D),F) )
                           => ( r1_xreal_0(B,E)
                              | r1_xreal_0(B,F)
                              | ( C = D
                                & E = F ) ) ) ) ) ) ) ) ) )).

fof(d1_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k3_nat_1(A,B)
              <=> ~ ( ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ~ ( A = k2_xcmplx_0(k3_xcmplx_0(B,C),D)
                            & ~ r1_xreal_0(B,D) ) )
                    & ~ ( C = np__0
                        & B = np__0 ) ) ) ) ) ) )).

fof(d2_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k4_nat_1(A,B)
              <=> ~ ( ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ~ ( A = k2_xcmplx_0(k3_xcmplx_0(B,D),C)
                            & ~ r1_xreal_0(B,C) ) )
                    & ~ ( C = np__0
                        & B = np__0 ) ) ) ) ) ) )).

fof(t44_nat_1,axiom,(
    $true )).

fof(t45_nat_1,axiom,(
    $true )).

fof(t46_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(A,k4_nat_1(B,A)) ) ) ) )).

fof(t47_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,np__0)
           => B = k2_xcmplx_0(k3_xcmplx_0(A,k3_nat_1(B,A)),k4_nat_1(B,A)) ) ) ) )).

fof(d3_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_nat_1(A,B)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k3_xcmplx_0(A,C) ) ) ) ) )).

fof(t48_nat_1,axiom,(
    $true )).

fof(t49_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_nat_1(A,B)
          <=> B = k3_xcmplx_0(A,k3_nat_1(B,A)) ) ) ) )).

fof(t50_nat_1,axiom,(
    $true )).

fof(t51_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_nat_1(A,B)
                  & r1_nat_1(B,C) )
               => r1_nat_1(A,C) ) ) ) ) )).

fof(t52_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( r1_nat_1(A,B)
              & r1_nat_1(B,A) )
           => A = B ) ) ) )).

fof(t53_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( r1_nat_1(A,np__0)
        & r1_nat_1(np__1,A) ) ) )).

fof(t54_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_nat_1(B,A)
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(B,A) ) ) ) ) )).

fof(t55_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_nat_1(A,B)
                  & r1_nat_1(A,C) )
               => r1_nat_1(A,k2_xcmplx_0(B,C)) ) ) ) ) )).

fof(t56_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( r1_nat_1(A,B)
               => r1_nat_1(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t57_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_nat_1(A,B)
                  & r1_nat_1(A,k2_xcmplx_0(B,C)) )
               => r1_nat_1(A,C) ) ) ) ) )).

fof(t58_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_nat_1(A,B)
                  & r1_nat_1(A,C) )
               => r1_nat_1(A,k4_nat_1(B,C)) ) ) ) ) )).

fof(d4_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k5_nat_1(A,B)
              <=> ( r1_nat_1(A,C)
                  & r1_nat_1(B,C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_nat_1(A,D)
                          & r1_nat_1(B,D) )
                       => r1_nat_1(C,D) ) ) ) ) ) ) ) )).

fof(d5_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k6_nat_1(A,B)
              <=> ( r1_nat_1(C,A)
                  & r1_nat_1(C,B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_nat_1(D,A)
                          & r1_nat_1(D,B) )
                       => r1_nat_1(D,C) ) ) ) ) ) ) ) )).

fof(t59_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(A,B),A)
          <=> r1_xreal_0(np__1,B) ) ) ) )).

fof(t60_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(B,A)
           => m2_subset_1(k6_xcmplx_0(B,np__1),k1_numbers,k5_numbers) ) ) ) )).

fof(t61_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => m2_subset_1(k6_xcmplx_0(B,A),k1_numbers,k5_numbers) ) ) ) )).

fof(t62_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( k4_nat_1(A,np__2) = np__0
        | k4_nat_1(A,np__2) = np__1 ) ) )).

fof(t63_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k4_nat_1(k3_xcmplx_0(A,B),A) = np__0 ) ) )).

fof(t64_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,np__1)
           => k4_nat_1(np__1,A) = np__1 ) ) ) )).

fof(t65_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ( ( k4_nat_1(A,C) = np__0
                      & B = k6_xcmplx_0(A,k3_xcmplx_0(D,C)) )
                   => k4_nat_1(B,C) = np__0 ) ) ) ) ) )).

fof(t66_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( k4_nat_1(A,C) = np__0
               => ( C = np__0
                  | r1_xreal_0(C,B)
                  | k4_nat_1(k2_xcmplx_0(A,B),C) = B ) ) ) ) ) )).

fof(t67_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( k4_nat_1(A,C) = np__0
               => k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(B,C) ) ) ) ) )).

fof(t68_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( A != np__0
           => k3_nat_1(k3_xcmplx_0(A,B),A) = B ) ) ) )).

fof(t69_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( k4_nat_1(A,B) = np__0
               => k3_nat_1(k2_xcmplx_0(A,C),B) = k1_nat_1(k3_nat_1(A,B),k3_nat_1(C,B)) ) ) ) ) )).

fof(t70_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(k1_nat_1(B,np__1),A)
              & r1_xreal_0(B,A)
              & A != B ) ) ) )).

fof(t71_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(np__2,A)
          & A != np__0
          & A != np__1 ) ) )).

fof(t72_nat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B != np__0
           => k3_nat_1(k2_nat_1(A,B),B) = A ) ) ) )).

fof(t73_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k4_nat_1(A,B) = k4_nat_1(k2_xcmplx_0(k3_xcmplx_0(B,C),A),B) ) ) ) )).

fof(t74_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(k2_xcmplx_0(k4_nat_1(A,C),B),C) ) ) ) )).

fof(t75_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(k2_xcmplx_0(A,k4_nat_1(B,C)),C) ) ) ) )).

fof(t76_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,B)
           => k4_nat_1(B,A) = B ) ) ) )).

fof(t77_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k4_nat_1(A,A) = np__0 ) )).

fof(t78_nat_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => np__0 = k4_nat_1(np__0,A) ) )).

fof(s1_nat_1,axiom,
    ( ( p1_s1_nat_1(np__0)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_nat_1(A)
           => p1_s1_nat_1(k1_nat_1(A,np__1)) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_nat_1(A) ) )).

fof(s2_nat_1,axiom,
    ( ( p1_s2_nat_1(np__0)
      & ! [A] :
          ( v4_ordinal2(A)
         => ( p1_s2_nat_1(A)
           => p1_s2_nat_1(k2_xcmplx_0(A,np__1)) ) ) )
   => ! [A] :
        ( v4_ordinal2(A)
       => p1_s2_nat_1(A) ) )).

fof(s3_nat_1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p1_s3_nat_1(A,B)
            <=> ~ ( ~ ( A = np__0
                      & B = f1_s3_nat_1 )
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ~ ( A = k1_nat_1(C,np__1)
                              & p1_s3_nat_1(C,D)
                              & B = f2_s3_nat_1(A,D) ) ) ) ) ) ) )
   => ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ? [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
              & p1_s3_nat_1(A,B) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( p1_s3_nat_1(A,B)
                      & p1_s3_nat_1(A,C) )
                   => B = C ) ) ) ) ) )).

fof(s4_nat_1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(A,B)
               => p1_s4_nat_1(B) ) )
         => p1_s4_nat_1(A) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s4_nat_1(A) ) )).

fof(s5_nat_1,axiom,
    ( ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s5_nat_1(A) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s5_nat_1(A)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p1_s5_nat_1(B)
             => r1_xreal_0(A,B) ) ) ) )).

fof(s6_nat_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s6_nat_1(A)
           => r1_xreal_0(A,f1_s6_nat_1) ) )
      & ? [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
          & p1_s6_nat_1(A) ) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s6_nat_1(A)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p1_s6_nat_1(B)
             => r1_xreal_0(B,A) ) ) ) )).

fof(s7_nat_1,axiom,
    ( ( ? [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
          & p1_s7_nat_1(A) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ~ ( A != np__0
              & p1_s7_nat_1(A)
              & ! [B] :
                  ( m2_subset_1(B,k1_numbers,k5_numbers)
                 => ~ ( ~ r1_xreal_0(A,B)
                      & p1_s7_nat_1(B) ) ) ) ) )
   => p1_s7_nat_1(np__0) )).

fof(s8_nat_1,axiom,
    ( ( ~ r1_xreal_0(f3_s8_nat_1,np__0)
      & ~ r1_xreal_0(f2_s8_nat_1,f3_s8_nat_1)
      & f1_s8_nat_1(np__0) = f2_s8_nat_1
      & f1_s8_nat_1(np__1) = f3_s8_nat_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => f1_s8_nat_1(k1_nat_1(A,np__2)) = k4_nat_1(f1_s8_nat_1(A),f1_s8_nat_1(k1_nat_1(A,np__1))) ) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & f1_s8_nat_1(A) = k6_nat_1(f2_s8_nat_1,f3_s8_nat_1)
        & f1_s8_nat_1(k1_nat_1(A,np__1)) = np__0 ) )).

fof(reflexivity_r1_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => r1_nat_1(A,A) ) )).

fof(dt_k1_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(commutativity_k1_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_nat_1(A,B) = k1_nat_1(B,A) ) )).

fof(redefinition_k1_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) )).

fof(dt_k2_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(commutativity_k2_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k2_nat_1(A,B) = k2_nat_1(B,A) ) )).

fof(redefinition_k2_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) )).

fof(dt_k3_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m2_subset_1(k3_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k4_nat_1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m2_subset_1(k4_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k5_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k5_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(commutativity_k5_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k5_nat_1(A,B) = k5_nat_1(B,A) ) )).

fof(idempotence_k5_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k5_nat_1(A,A) = A ) )).

fof(dt_k6_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k6_nat_1(A,B),k1_numbers,k5_numbers) ) )).

fof(commutativity_k6_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k6_nat_1(A,B) = k6_nat_1(B,A) ) )).

fof(idempotence_k6_nat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k6_nat_1(A,A) = A ) )).
%------------------------------------------------------------------------------