SET007 Axioms: SET007+49.ax


%------------------------------------------------------------------------------
% File     : SET007+49 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Rational 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    : rat_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  117 (  45 unit)
%            Number of atoms       :  379 (  75 equality)
%            Maximal formula depth :   18 (   4 average)
%            Number of connectives :  326 (  64 ~  ;   7  |; 132  &)
%                                         (  14 <=>; 109 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   10 (   1 propositional; 0-3 arity)
%            Number of functors    :   16 (   5 constant; 0-2 arity)
%            Number of variables   :  132 (   0 singleton; 117 !;  15 ?)
%            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_rat_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_rat_1(A) ) )).

fof(rc2_rat_1,axiom,(
    ? [A] : v1_rat_1(A) )).

fof(cc1_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) )).

fof(cc2_rat_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v1_rat_1(A) ) ) )).

fof(fc1_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc2_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc3_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_rat_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc4_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc5_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_rat_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc6_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc7_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc8_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_rat_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc9_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc10_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc11_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_rat_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc12_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc13_rat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_rat_1(k2_xcmplx_0(A,B)) ) ) )).

fof(fc14_rat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_rat_1(k6_xcmplx_0(A,B)) ) ) )).

fof(fc15_rat_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_rat_1(k3_xcmplx_0(A,B)) ) ) )).

fof(fc16_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & v1_rat_1(k4_xcmplx_0(A)) ) ) )).

fof(fc17_rat_1,axiom,(
    ! [A,B] :
      ( ( v1_rat_1(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(A,B))
        & v1_xreal_0(k7_xcmplx_0(A,B))
        & v1_rat_1(k7_xcmplx_0(A,B)) ) ) )).

fof(fc18_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( v1_xcmplx_0(k5_xcmplx_0(A))
        & v1_xreal_0(k5_xcmplx_0(A))
        & v1_rat_1(k5_xcmplx_0(A)) ) ) )).

fof(d1_rat_1,axiom,(
    ! [A] :
      ( A = k3_numbers
    <=> ! [B] :
          ( r2_hidden(B,A)
        <=> ? [C] :
              ( v1_int_1(C)
              & ? [D] :
                  ( v1_int_1(D)
                  & B = k7_xcmplx_0(C,D) ) ) ) ) )).

fof(d2_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
    <=> r2_hidden(A,k3_numbers) ) )).

fof(t1_rat_1,axiom,(
    ! [A] :
      ~ ( r2_hidden(A,k3_numbers)
        & ! [B] :
            ( v1_int_1(B)
           => ! [C] :
                ( v1_int_1(C)
               => ~ ( C != np__0
                    & A = k7_xcmplx_0(B,C) ) ) ) ) )).

fof(t2_rat_1,axiom,(
    $true )).

fof(t3_rat_1,axiom,(
    ! [A] :
      ~ ( v1_rat_1(A)
        & ! [B] :
            ( v1_int_1(B)
           => ! [C] :
                ( v1_int_1(C)
               => ~ ( C != np__0
                    & A = k7_xcmplx_0(B,C) ) ) ) ) )).

fof(t4_rat_1,axiom,(
    $true )).

fof(t5_rat_1,axiom,(
    $true )).

fof(t6_rat_1,axiom,(
    ! [A] :
      ( ? [B] :
          ( v1_int_1(B)
          & ? [C] :
              ( v1_int_1(C)
              & A = k7_xcmplx_0(B,C) ) )
     => v1_rat_1(A) ) )).

fof(t7_rat_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => v1_rat_1(A) ) )).

fof(t8_rat_1,axiom,(
    $true )).

fof(t9_rat_1,axiom,(
    $true )).

fof(t10_rat_1,axiom,(
    $true )).

fof(t11_rat_1,axiom,(
    $true )).

fof(t12_rat_1,axiom,(
    $true )).

fof(t13_rat_1,axiom,(
    $true )).

fof(t14_rat_1,axiom,(
    $true )).

fof(t15_rat_1,axiom,(
    $true )).

fof(t16_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => v1_rat_1(k7_xcmplx_0(A,B)) ) ) )).

fof(t17_rat_1,axiom,(
    $true )).

fof(t18_rat_1,axiom,(
    $true )).

fof(t19_rat_1,axiom,(
    $true )).

fof(t20_rat_1,axiom,(
    $true )).

fof(t21_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => v1_rat_1(k5_xcmplx_0(A)) ) )).

fof(t22_rat_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(B,A)
              & ! [C] :
                  ( v1_rat_1(C)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t23_rat_1,axiom,(
    $true )).

fof(t24_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ? [B] :
          ( v1_int_1(B)
          & ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & C != np__0
              & A = k7_xcmplx_0(B,C) ) ) ) )).

fof(t25_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ? [B] :
          ( v1_int_1(B)
          & ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & C != np__0
              & A = k7_xcmplx_0(B,C)
              & ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( A = k7_xcmplx_0(D,E)
                       => ( E = np__0
                          | r1_xreal_0(C,E) ) ) ) ) ) ) ) )).

fof(d3_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k1_rat_1(A)
          <=> ( B != np__0
              & ? [C] :
                  ( v1_int_1(C)
                  & A = k7_xcmplx_0(C,B) )
              & ! [C] :
                  ( v1_int_1(C)
                 => ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( A = k7_xcmplx_0(C,D)
                       => ( D = np__0
                          | r1_xreal_0(B,D) ) ) ) ) ) ) ) ) )).

fof(d4_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => k2_rat_1(A) = k3_xcmplx_0(k1_rat_1(A),A) ) )).

fof(t26_rat_1,axiom,(
    $true )).

fof(t27_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ~ r1_xreal_0(k1_rat_1(A),np__0) ) )).

fof(t28_rat_1,axiom,(
    $true )).

fof(t29_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => r1_xreal_0(np__1,k1_rat_1(A)) ) )).

fof(t30_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ~ r1_xreal_0(k2_real_1(k1_rat_1(A)),np__0) ) )).

fof(t31_rat_1,axiom,(
    $true )).

fof(t32_rat_1,axiom,(
    $true )).

fof(t33_rat_1,axiom,(
    $true )).

fof(t34_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => r1_xreal_0(k2_real_1(k1_rat_1(A)),np__1) ) )).

fof(t35_rat_1,axiom,(
    $true )).

fof(t36_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k2_rat_1(A) = np__0
      <=> A = np__0 ) ) )).

fof(t37_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( A = k7_xcmplx_0(k2_rat_1(A),k1_rat_1(A))
        & A = k3_xcmplx_0(k2_rat_1(A),k2_real_1(k1_rat_1(A)))
        & A = k3_xcmplx_0(k2_real_1(k1_rat_1(A)),k2_rat_1(A)) ) ) )).

fof(t38_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( A != np__0
       => k1_rat_1(A) = k7_xcmplx_0(k2_rat_1(A),A) ) ) )).

fof(t39_rat_1,axiom,(
    $true )).

fof(t40_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( v1_int_1(A)
       => ( k1_rat_1(A) = np__1
          & k2_rat_1(A) = A ) ) ) )).

fof(t41_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ( k2_rat_1(A) = A
          | k1_rat_1(A) = np__1 )
       => v1_int_1(A) ) ) )).

fof(t42_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k2_rat_1(A) = A
      <=> k1_rat_1(A) = np__1 ) ) )).

fof(t43_rat_1,axiom,(
    $true )).

fof(t44_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( r1_xreal_0(np__0,A)
       => ( ( k2_rat_1(A) != A
            & k1_rat_1(A) != np__1 )
          | m2_subset_1(A,k1_numbers,k5_numbers) ) ) ) )).

fof(t45_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(k1_rat_1(A),np__1)
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(k1_rat_1(A),np__1) ) ) ) )).

fof(t46_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(np__1,k2_real_1(k1_rat_1(A)))
            & v1_int_1(A) )
        & ~ ( ~ v1_int_1(A)
            & r1_xreal_0(np__1,k2_real_1(k1_rat_1(A))) ) ) ) )).

fof(t47_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k2_rat_1(A) = k1_rat_1(A)
      <=> A = np__1 ) ) )).

fof(t48_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k2_rat_1(A) = k1_real_1(k1_rat_1(A))
      <=> A = k1_real_1(np__1) ) ) )).

fof(t49_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k4_xcmplx_0(k2_rat_1(A)) = k1_rat_1(A)
      <=> A = k1_real_1(np__1) ) ) )).

fof(t50_rat_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( A != np__0
           => B = k7_xcmplx_0(k3_xcmplx_0(k2_rat_1(B),A),k3_xcmplx_0(k1_rat_1(B),A)) ) ) ) )).

fof(t51_rat_1,axiom,(
    $true )).

fof(t52_rat_1,axiom,(
    $true )).

fof(t53_rat_1,axiom,(
    $true )).

fof(t54_rat_1,axiom,(
    $true )).

fof(t55_rat_1,axiom,(
    $true )).

fof(t56_rat_1,axiom,(
    $true )).

fof(t57_rat_1,axiom,(
    $true )).

fof(t58_rat_1,axiom,(
    $true )).

fof(t59_rat_1,axiom,(
    $true )).

fof(t60_rat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ~ ( A != np__0
                  & C = k7_xcmplx_0(B,A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( B = k3_xcmplx_0(k2_rat_1(C),D)
                          & A = k2_nat_1(k1_rat_1(C),D) ) ) ) ) ) ) )).

fof(t61_rat_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ~ ( C = k7_xcmplx_0(A,B)
                  & B != np__0
                  & ! [D] :
                      ( v1_int_1(D)
                     => ~ ( A = k3_xcmplx_0(k2_rat_1(C),D)
                          & B = k3_xcmplx_0(k1_rat_1(C),D) ) ) ) ) ) ) )).

fof(t62_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(B,np__1)
              & ? [C] :
                  ( v1_int_1(C)
                  & ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & k2_rat_1(A) = k3_xcmplx_0(C,B)
                      & k1_rat_1(A) = k2_nat_1(D,B) ) ) ) ) ) )).

fof(t63_rat_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ( C = k7_xcmplx_0(B,A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__1)
                          & ? [E] :
                              ( v1_int_1(E)
                              & ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & B = k3_xcmplx_0(E,D)
                                  & A = k2_nat_1(F,D) ) ) ) ) )
               => ( A = np__0
                  | ( A = k1_rat_1(C)
                    & B = k2_rat_1(C) ) ) ) ) ) ) )).

fof(t64_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(k1_real_1(np__1),A)
            & r1_xreal_0(k1_real_1(k1_rat_1(A)),k2_rat_1(A)) )
        & ~ ( ~ r1_xreal_0(k1_real_1(k1_rat_1(A)),k2_rat_1(A))
            & r1_xreal_0(k1_real_1(np__1),A) ) ) ) )).

fof(t65_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( r1_xreal_0(A,k1_real_1(np__1))
      <=> r1_xreal_0(k2_rat_1(A),k1_real_1(k1_rat_1(A))) ) ) )).

fof(t66_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(k1_real_1(np__1),A)
            & r1_xreal_0(k4_xcmplx_0(k2_rat_1(A)),k1_rat_1(A)) )
        & ~ ( ~ r1_xreal_0(k4_xcmplx_0(k2_rat_1(A)),k1_rat_1(A))
            & r1_xreal_0(k1_real_1(np__1),A) ) ) ) )).

fof(t67_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( r1_xreal_0(A,k1_real_1(np__1))
      <=> r1_xreal_0(k1_rat_1(A),k4_xcmplx_0(k2_rat_1(A))) ) ) )).

fof(t68_rat_1,axiom,(
    $true )).

fof(t69_rat_1,axiom,(
    $true )).

fof(t70_rat_1,axiom,(
    $true )).

fof(t71_rat_1,axiom,(
    $true )).

fof(t72_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(np__1,A)
            & r1_xreal_0(k1_rat_1(A),k2_rat_1(A)) )
        & ~ ( ~ r1_xreal_0(k1_rat_1(A),k2_rat_1(A))
            & r1_xreal_0(np__1,A) ) ) ) )).

fof(t73_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( r1_xreal_0(A,np__1)
      <=> r1_xreal_0(k2_rat_1(A),k1_rat_1(A)) ) ) )).

fof(t74_rat_1,axiom,(
    $true )).

fof(t75_rat_1,axiom,(
    $true )).

fof(t76_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( ~ ( ~ r1_xreal_0(np__0,A)
            & r1_xreal_0(np__0,k2_rat_1(A)) )
        & ~ ( ~ r1_xreal_0(np__0,k2_rat_1(A))
            & r1_xreal_0(np__0,A) ) ) ) )).

fof(t77_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( r1_xreal_0(A,np__0)
      <=> r1_xreal_0(k2_rat_1(A),np__0) ) ) )).

fof(t78_rat_1,axiom,(
    $true )).

fof(t79_rat_1,axiom,(
    $true )).

fof(t80_rat_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ ( ~ r1_xreal_0(B,A)
                & r1_xreal_0(k2_rat_1(B),k3_xcmplx_0(A,k1_rat_1(B))) )
            & ~ ( ~ r1_xreal_0(k2_rat_1(B),k3_xcmplx_0(A,k1_rat_1(B)))
                & r1_xreal_0(B,A) ) ) ) ) )).

fof(t81_rat_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( r1_xreal_0(A,B)
          <=> r1_xreal_0(k3_xcmplx_0(A,k1_rat_1(B)),k2_rat_1(B)) ) ) ) )).

fof(t82_rat_1,axiom,(
    $true )).

fof(t83_rat_1,axiom,(
    $true )).

fof(t84_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( k1_rat_1(A) = k1_rat_1(B)
              & k2_rat_1(A) = k2_rat_1(B) )
           => A = B ) ) ) )).

fof(t85_rat_1,axiom,(
    $true )).

fof(t86_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ ( ~ r1_xreal_0(B,A)
                & r1_xreal_0(k3_xcmplx_0(k2_rat_1(B),k1_rat_1(A)),k3_xcmplx_0(k2_rat_1(A),k1_rat_1(B))) )
            & ~ ( ~ r1_xreal_0(k3_xcmplx_0(k2_rat_1(B),k1_rat_1(A)),k3_xcmplx_0(k2_rat_1(A),k1_rat_1(B)))
                & r1_xreal_0(B,A) ) ) ) ) )).

fof(t87_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ( k1_rat_1(k4_xcmplx_0(A)) = k1_rat_1(A)
        & k2_rat_1(k4_xcmplx_0(A)) = k4_xcmplx_0(k2_rat_1(A)) ) ) )).

fof(t88_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( B = k7_xcmplx_0(np__1,A)
             => ( r1_xreal_0(A,np__0)
                | ( k2_rat_1(B) = k1_rat_1(A)
                  & k1_rat_1(B) = k2_rat_1(A) ) ) )
            & ( ( k2_rat_1(B) = k1_rat_1(A)
                & k1_rat_1(B) = k2_rat_1(A) )
             => ( ~ r1_xreal_0(A,np__0)
                & B = k7_xcmplx_0(np__1,A) ) ) ) ) ) )).

fof(t89_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( B = k7_xcmplx_0(np__1,A)
             => ( r1_xreal_0(np__0,A)
                | ( k2_rat_1(B) = k1_real_1(k1_rat_1(A))
                  & k1_rat_1(B) = k4_xcmplx_0(k2_rat_1(A)) ) ) )
            & ( ( k2_rat_1(B) = k1_real_1(k1_rat_1(A))
                & k1_rat_1(B) = k4_xcmplx_0(k2_rat_1(A)) )
             => ( ~ r1_xreal_0(np__0,A)
                & B = k7_xcmplx_0(np__1,A) ) ) ) ) ) )).

fof(dt_k1_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => m2_subset_1(k1_rat_1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k2_rat_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => v1_int_1(k2_rat_1(A)) ) )).
%------------------------------------------------------------------------------