SET007 Axioms: SET007+83.ax


%------------------------------------------------------------------------------
% File     : SET007+83 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : sigma-Fields and Probability
% 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    : prob_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  132 (  32 unit)
%            Number of atoms       :  572 (  54 equality)
%            Maximal formula depth :   16 (   6 average)
%            Number of connectives :  487 (  47 ~  ;   0  |; 199  &)
%                                         (  15 <=>; 226 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   1 propositional; 0-3 arity)
%            Number of functors    :   44 (   8 constant; 0-4 arity)
%            Number of variables   :  339 (   0 singleton; 320 !;  19 ?)
%            Maximal term depth    :    5 (   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_prob_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & ~ v1_xboole_0(B)
      & v2_finsub_1(B)
      & v1_prob_1(B,A) ) )).

fof(cc1_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ( v2_finsub_1(B)
        & v1_prob_1(B,A) ) ) )).

fof(rc2_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ? [C] :
          ( m3_prob_1(C,A,B)
          & v1_xboole_0(C)
          & v1_funct_1(C)
          & v1_membered(C)
          & v2_membered(C)
          & v3_membered(C)
          & v4_membered(C)
          & v5_membered(C) ) ) )).

fof(t1_prob_1,axiom,(
    $true )).

fof(t2_prob_1,axiom,(
    $true )).

fof(t3_prob_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(C,D)
                     => k2_seq_1(k5_numbers,k1_numbers,B,D) = A ) ) )
           => ( v4_seq_2(B)
              & k2_seq_2(B) = A ) ) ) ) )).

fof(d1_prob_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v1_prob_1(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ( r2_hidden(C,B)
             => r2_hidden(k3_subset_1(A,C),B) ) ) ) ) )).

fof(t4_prob_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => m1_subset_1(k2_tarski(B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(t5_prob_1,axiom,(
    $true )).

fof(t6_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
          & r2_hidden(C,B) ) ) )).

fof(t7_prob_1,axiom,(
    $true )).

fof(t8_prob_1,axiom,(
    $true )).

fof(t9_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C,D] :
          ( ( r2_hidden(C,B)
            & r2_hidden(D,B) )
         => r2_hidden(k2_xboole_0(C,D),B) ) ) )).

fof(t10_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => r2_hidden(k1_xboole_0,B) ) )).

fof(t11_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => r2_hidden(A,B) ) )).

fof(t12_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( r2_hidden(C,B)
                  & r2_hidden(D,B) )
               => r2_hidden(k6_subset_1(A,C,D),B) ) ) ) ) )).

fof(t13_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C,D] :
          ( r1_xboole_0(k4_xboole_0(C,D),D)
          & ( ( r2_hidden(C,B)
              & r2_hidden(D,B) )
           => r2_hidden(k2_xboole_0(k4_xboole_0(C,D),D),B) ) ) ) )).

fof(t14_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k2_tarski(k1_xboole_0,A))
      & v2_finsub_1(k2_tarski(k1_xboole_0,A))
      & v1_prob_1(k2_tarski(k1_xboole_0,A),A)
      & m1_subset_1(k2_tarski(k1_xboole_0,A),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(t15_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v2_finsub_1(k1_zfmisc_1(A))
      & v1_prob_1(k1_zfmisc_1(A),A)
      & m1_subset_1(k1_zfmisc_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(t16_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ( r1_tarski(k2_tarski(k1_xboole_0,A),B)
        & r1_tarski(B,k1_zfmisc_1(A)) ) ) )).

fof(t17_prob_1,axiom,(
    $true )).

fof(t18_prob_1,axiom,(
    ! [A] :
      ( ! [B] :
          ~ ( r2_hidden(B,k2_zfmisc_1(k5_numbers,k1_tarski(A)))
            & ! [C,D] : k4_tarski(C,D) != B )
      & ! [B,C,D] :
          ( ( r2_hidden(k4_tarski(B,C),k2_zfmisc_1(k5_numbers,k1_tarski(A)))
            & r2_hidden(k4_tarski(B,D),k2_zfmisc_1(k5_numbers,k1_tarski(A))) )
         => C = D ) ) )).

fof(t19_prob_1,axiom,(
    ! [A] :
    ? [B] :
      ( v1_relat_1(B)
      & v1_funct_1(B)
      & k1_relat_1(B) = k5_numbers
      & ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => k1_funct_1(B,C) = A ) ) )).

fof(t20_prob_1,axiom,(
    $true )).

fof(t21_prob_1,axiom,(
    ! [A] :
    ? [B] :
      ( v1_funct_1(B)
      & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
      & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A))
      & ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,k1_zfmisc_1(A),B,C) = A ) ) )).

fof(t22_prob_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k5_numbers,k1_zfmisc_1(A))
              & m2_relset_1(D,k5_numbers,k1_zfmisc_1(A))
              & k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,np__0) = B
              & ! [E] :
                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                 => ( E != np__0
                   => k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,E) = C ) ) ) ) ) )).

fof(t23_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => m1_subset_1(k3_tarski(k2_relat_1(B)),k1_zfmisc_1(A)) ) )).

fof(t24_prob_1,axiom,(
    $true )).

fof(t25_prob_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
     => ( r2_hidden(B,k2_prob_1(A,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & r2_hidden(B,k1_prob_1(A,C,D)) ) ) ) )).

fof(t26_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ? [C] :
          ( v1_funct_1(C)
          & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
          & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A))
          & ! [D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => k1_prob_1(A,C,D) = k3_subset_1(A,k1_prob_1(A,B,D)) ) ) ) )).

fof(d2_prob_1,axiom,(
    $true )).

fof(d3_prob_1,axiom,(
    $true )).

fof(d4_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
         => ( C = k3_prob_1(A,B)
          <=> ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => k1_prob_1(A,C,D) = k3_subset_1(A,k1_prob_1(A,B,D)) ) ) ) ) )).

fof(d5_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => k4_prob_1(A,B) = k3_subset_1(A,k2_prob_1(A,k3_prob_1(A,B))) ) )).

fof(t27_prob_1,axiom,(
    $true )).

fof(t28_prob_1,axiom,(
    $true )).

fof(t29_prob_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
     => ( r2_hidden(B,k4_prob_1(A,C))
      <=> ! [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
           => r2_hidden(B,k1_prob_1(A,C,D)) ) ) ) )).

fof(t30_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( k1_prob_1(A,B,np__0) = C
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( E != np__0
                       => k1_prob_1(A,B,E) = D ) ) )
               => k4_prob_1(A,B) = k5_subset_1(A,C,D) ) ) ) ) )).

fof(d6_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v2_prob_1(B,A)
      <=> ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => ( r1_xreal_0(C,D)
                 => r1_tarski(k1_prob_1(A,B,D),k1_prob_1(A,B,C)) ) ) ) ) ) )).

fof(d7_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v3_prob_1(B,A)
      <=> ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => ( r1_xreal_0(C,D)
                 => r1_tarski(k1_prob_1(A,B,C),k1_prob_1(A,B,D)) ) ) ) ) ) )).

fof(d8_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ( m1_prob_1(B,A)
      <=> ( ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r2_hidden(k1_prob_1(A,C,D),B) )
               => r2_hidden(k4_prob_1(A,C),B) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r2_hidden(C,B)
               => r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ) )).

fof(t31_prob_1,axiom,(
    $true )).

fof(t32_prob_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( m1_prob_1(B,A)
      <=> ( r1_tarski(B,k1_zfmisc_1(A))
          & ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r2_hidden(k1_prob_1(A,C,D),B) )
               => r2_hidden(k4_prob_1(A,C),B) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r2_hidden(C,B)
               => r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ) )).

fof(t33_prob_1,axiom,(
    $true )).

fof(t34_prob_1,axiom,(
    $true )).

fof(t35_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(A,B)
     => ( ~ v1_xboole_0(A)
        & v2_finsub_1(A)
        & v1_prob_1(A,B)
        & m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(B))) ) ) )).

fof(t36_prob_1,axiom,(
    $true )).

fof(t37_prob_1,axiom,(
    $true )).

fof(t38_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
          & r2_hidden(C,B) ) ) )).

fof(t39_prob_1,axiom,(
    $true )).

fof(t40_prob_1,axiom,(
    $true )).

fof(t41_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( r2_hidden(C,B)
                  & r2_hidden(D,B) )
               => r2_hidden(k4_subset_1(A,C,D),B) ) ) ) ) )).

fof(t42_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => r2_hidden(k1_xboole_0,B) ) )).

fof(t43_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => r2_hidden(A,B) ) )).

fof(t44_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( r2_hidden(C,B)
                  & r2_hidden(D,B) )
               => r2_hidden(k6_subset_1(A,C,D),B) ) ) ) ) )).

fof(d9_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
         => ( m2_prob_1(C,A,B)
          <=> ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => r2_hidden(k1_prob_1(A,C,D),B) ) ) ) ) )).

fof(t45_prob_1,axiom,(
    $true )).

fof(t46_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m2_prob_1(C,A,B)
         => r2_hidden(k2_prob_1(A,C),B) ) ) )).

fof(d10_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( m3_prob_1(C,A,B)
          <=> r2_hidden(C,B) ) ) ) )).

fof(t47_prob_1,axiom,(
    $true )).

fof(t48_prob_1,axiom,(
    ! [A,B,C] :
      ( m1_prob_1(C,A)
     => ( r2_hidden(B,C)
       => m3_prob_1(B,A,C) ) ) )).

fof(t49_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m3_prob_1(C,A,B)
         => ! [D] :
              ( m3_prob_1(D,A,B)
             => m3_prob_1(k5_subset_1(A,C,D),A,B) ) ) ) )).

fof(t50_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m3_prob_1(C,A,B)
         => m3_prob_1(k3_subset_1(A,C),A,B) ) ) )).

fof(t51_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m3_prob_1(C,A,B)
         => ! [D] :
              ( m3_prob_1(D,A,B)
             => m3_prob_1(k4_subset_1(A,C,D),A,B) ) ) ) )).

fof(t52_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => m3_prob_1(k1_xboole_0,A,B) ) )).

fof(t53_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => m3_prob_1(A,A,B) ) )).

fof(t54_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m3_prob_1(C,A,B)
         => ! [D] :
              ( m3_prob_1(D,A,B)
             => m3_prob_1(k6_subset_1(A,C,D),A,B) ) ) ) )).

fof(d11_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => k5_prob_1(A,B) = A ) )).

fof(t55_prob_1,axiom,(
    $true )).

fof(t56_prob_1,axiom,(
    $true )).

fof(t57_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m1_prob_1(C,A)
             => ( m2_prob_1(B,A,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => m3_prob_1(k1_prob_1(A,B,D),A,C) ) ) ) ) ) )).

fof(t58_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m1_prob_1(C,A)
             => ( m2_prob_1(B,A,C)
               => m3_prob_1(k2_prob_1(A,B),A,C) ) ) ) ) )).

fof(t59_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_prob_1(C,A)
         => ? [D] :
              ( v1_relat_1(D)
              & v1_funct_1(D)
              & k1_relat_1(D) = C
              & ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(A))
                 => ( r2_hidden(E,C)
                   => ( ( r2_hidden(B,E)
                       => k1_funct_1(D,E) = np__1 )
                      & ( ~ r2_hidden(B,E)
                       => k1_funct_1(D,E) = np__0 ) ) ) ) ) ) ) )).

fof(t60_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_prob_1(C,A)
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,C,k1_numbers)
              & m2_relset_1(D,C,k1_numbers)
              & ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(A))
                 => ( r2_hidden(E,C)
                   => ( ( r2_hidden(B,E)
                       => k2_seq_1(C,k1_numbers,D,E) = np__1 )
                      & ( ~ r2_hidden(B,E)
                       => k2_seq_1(C,k1_numbers,D,E) = np__0 ) ) ) ) ) ) ) )).

fof(t61_prob_1,axiom,(
    $true )).

fof(t62_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,B,k1_numbers)
                    & m2_relset_1(D,B,k1_numbers) )
                 => ( v1_funct_1(k5_relat_1(C,D))
                    & v1_funct_2(k5_relat_1(C,D),k5_numbers,k1_numbers)
                    & m2_relset_1(k5_relat_1(C,D),k5_numbers,k1_numbers) ) ) ) ) ) )).

fof(d12_prob_1,axiom,(
    $true )).

fof(d13_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ( m4_prob_1(C,A,B)
              <=> ( ! [D] :
                      ( m3_prob_1(D,A,B)
                     => r1_xreal_0(np__0,k10_prob_1(A,B,C,D)) )
                  & k2_seq_1(B,k1_numbers,C,A) = np__1
                  & ! [D] :
                      ( m3_prob_1(D,A,B)
                     => ! [E] :
                          ( m3_prob_1(E,A,B)
                         => ( r1_xboole_0(D,E)
                           => k10_prob_1(A,B,C,k7_prob_1(A,B,D,E)) = k2_xcmplx_0(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) )
                  & ! [D] :
                      ( m2_prob_1(D,A,B)
                     => ( v2_prob_1(D,A)
                       => ( v4_seq_2(k9_prob_1(A,B,D,C))
                          & k2_seq_2(k9_prob_1(A,B,D,C)) = k2_seq_1(B,k1_numbers,C,k4_prob_1(A,D)) ) ) ) ) ) ) ) ) )).

fof(t63_prob_1,axiom,(
    $true )).

fof(t64_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => k2_seq_1(B,k1_numbers,C,k1_xboole_0) = np__0 ) ) ) )).

fof(t65_prob_1,axiom,(
    $true )).

fof(t66_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => k10_prob_1(A,B,C,k5_prob_1(A,B)) = np__1 ) ) ) )).

fof(t67_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m4_prob_1(D,A,B)
                 => k2_xcmplx_0(k10_prob_1(A,B,D,k8_prob_1(A,B,k5_prob_1(A,B),C)),k10_prob_1(A,B,D,C)) = np__1 ) ) ) ) )).

fof(t68_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m4_prob_1(D,A,B)
                 => k10_prob_1(A,B,D,k8_prob_1(A,B,k5_prob_1(A,B),C)) = k6_xcmplx_0(np__1,k10_prob_1(A,B,D,C)) ) ) ) ) )).

fof(t69_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => ( r1_tarski(C,D)
                       => k10_prob_1(A,B,E,k8_prob_1(A,B,D,C)) = k6_xcmplx_0(k10_prob_1(A,B,E,D),k10_prob_1(A,B,E,C)) ) ) ) ) ) ) )).

fof(t70_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => ( r1_tarski(C,D)
                       => r1_xreal_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D)) ) ) ) ) ) ) )).

fof(t71_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m4_prob_1(D,A,B)
                 => r1_xreal_0(k10_prob_1(A,B,D,C),np__1) ) ) ) ) )).

fof(t72_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,k8_prob_1(A,B,D,C))) ) ) ) ) ) )).

fof(t73_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,k8_prob_1(A,B,D,k6_prob_1(A,B,C,D)))) ) ) ) ) ) )).

fof(t74_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)) = k6_xcmplx_0(k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D)),k10_prob_1(A,B,E,k6_prob_1(A,B,C,D))) ) ) ) ) ) )).

fof(t75_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => r1_xreal_0(k10_prob_1(A,B,E,k7_prob_1(A,B,C,D)),k2_xcmplx_0(k10_prob_1(A,B,E,C),k10_prob_1(A,B,E,D))) ) ) ) ) ) )).

fof(t76_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_prob_1(k1_zfmisc_1(A),A) ) )).

fof(d14_prob_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ! [C] :
              ( m1_prob_1(C,A)
             => ( C = k11_prob_1(A,B)
              <=> ( r1_tarski(B,C)
                  & ! [D] :
                      ( ( r1_tarski(B,D)
                        & m1_prob_1(D,A) )
                     => r1_tarski(C,D) ) ) ) ) ) ) )).

fof(d17_prob_1,axiom,(
    k14_prob_1 = k11_prob_1(k1_numbers,k13_prob_1) )).

fof(dt_m1_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(existence_m1_prob_1,axiom,(
    ! [A] :
    ? [B] : m1_prob_1(B,A) )).

fof(dt_m2_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m2_prob_1(C,A,B)
         => ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) ) ) ) )).

fof(existence_m2_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ? [C] : m2_prob_1(C,A,B) ) )).

fof(dt_m3_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ! [C] :
          ( m3_prob_1(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(A)) ) ) )).

fof(existence_m3_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => ? [C] : m3_prob_1(C,A,B) ) )).

fof(dt_m4_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A) )
     => ! [C] :
          ( m4_prob_1(C,A,B)
         => ( v1_funct_1(C)
            & v1_funct_2(C,B,k1_numbers)
            & m2_relset_1(C,B,k1_numbers) ) ) ) )).

fof(existence_m4_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A) )
     => ? [C] : m4_prob_1(C,A,B) ) )).

fof(dt_k1_prob_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k1_prob_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(redefinition_k1_prob_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => k1_prob_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k2_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => m1_subset_1(k2_prob_1(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k2_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => k2_prob_1(A,B) = k3_card_3(B) ) )).

fof(dt_k3_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k3_prob_1(A,B))
        & v1_funct_2(k3_prob_1(A,B),k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(k3_prob_1(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) )).

fof(involutiveness_k3_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => k3_prob_1(A,k3_prob_1(A,B)) = B ) )).

fof(dt_k4_prob_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => m1_subset_1(k4_prob_1(A,B),k1_zfmisc_1(A)) ) )).

fof(dt_k5_prob_1,axiom,(
    ! [A,B] :
      ( m1_prob_1(B,A)
     => m3_prob_1(k5_prob_1(A,B),A,B) ) )).

fof(dt_k6_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => m3_prob_1(k6_prob_1(A,B,C,D),A,B) ) )).

fof(commutativity_k6_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k6_prob_1(A,B,C,D) = k6_prob_1(A,B,D,C) ) )).

fof(idempotence_k6_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k6_prob_1(A,B,C,C) = C ) )).

fof(redefinition_k6_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k6_prob_1(A,B,C,D) = k3_xboole_0(C,D) ) )).

fof(dt_k7_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => m3_prob_1(k7_prob_1(A,B,C,D),A,B) ) )).

fof(commutativity_k7_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k7_prob_1(A,B,C,D) = k7_prob_1(A,B,D,C) ) )).

fof(idempotence_k7_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k7_prob_1(A,B,C,C) = C ) )).

fof(redefinition_k7_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k7_prob_1(A,B,C,D) = k2_xboole_0(C,D) ) )).

fof(dt_k8_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => m3_prob_1(k8_prob_1(A,B,C,D),A,B) ) )).

fof(redefinition_k8_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_prob_1(B,A)
        & m3_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => k8_prob_1(A,B,C,D) = k4_xboole_0(C,D) ) )).

fof(dt_k9_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B)
        & v1_funct_1(D)
        & v1_funct_2(D,B,k1_numbers)
        & m1_relset_1(D,B,k1_numbers) )
     => ( v1_funct_1(k9_prob_1(A,B,C,D))
        & v1_funct_2(k9_prob_1(A,B,C,D),k5_numbers,k1_numbers)
        & m2_relset_1(k9_prob_1(A,B,C,D),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k9_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B)
        & v1_funct_1(D)
        & v1_funct_2(D,B,k1_numbers)
        & m1_relset_1(D,B,k1_numbers) )
     => k9_prob_1(A,B,C,D) = k5_relat_1(C,D) ) )).

fof(dt_k10_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,B,k1_numbers)
        & m1_relset_1(C,B,k1_numbers)
        & m3_prob_1(D,A,B) )
     => m1_subset_1(k10_prob_1(A,B,C,D),k1_numbers) ) )).

fof(redefinition_k10_prob_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,B,k1_numbers)
        & m1_relset_1(C,B,k1_numbers)
        & m3_prob_1(D,A,B) )
     => k10_prob_1(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k11_prob_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => m1_prob_1(k11_prob_1(A,B),A) ) )).

fof(dt_k12_prob_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => m1_subset_1(k12_prob_1(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k13_prob_1,axiom,(
    m1_subset_1(k13_prob_1,k1_zfmisc_1(k1_zfmisc_1(k1_numbers))) )).

fof(dt_k14_prob_1,axiom,(
    m1_prob_1(k14_prob_1,k1_numbers) )).

fof(d15_prob_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k12_prob_1(A) = a_1_0_prob_1(A) ) )).

fof(d16_prob_1,axiom,(
    k13_prob_1 = a_0_0_prob_1 )).

fof(fraenkel_a_1_0_prob_1,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ( r2_hidden(A,a_1_0_prob_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & ~ r1_xreal_0(B,C) ) ) ) )).

fof(fraenkel_a_0_0_prob_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_prob_1)
    <=> ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
          & A = B
          & ? [C] :
              ( v1_xreal_0(C)
              & B = k12_prob_1(C) ) ) ) )).
%------------------------------------------------------------------------------