SET007 Axioms: SET007+364.ax


%------------------------------------------------------------------------------
% File     : SET007+364 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Coherent Space
% 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    : coh_sp [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  133 (  35 unit)
%            Number of atoms       :  508 ( 108 equality)
%            Maximal formula depth :   18 (   5 average)
%            Number of connectives :  409 (  34 ~  ;   2  |; 206  &)
%                                         (  32 <=>; 135 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-3 arity)
%            Number of functors    :   56 (   1 constant; 0-6 arity)
%            Number of variables   :  285 (   0 singleton; 258 !;  27 ?)
%            Maximal term depth    :    4 (   2 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_coh_sp,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_classes1(A)
      & v1_coh_sp(A) ) )).

fof(fc1_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k3_coh_sp(A)) )).

fof(cc1_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_coh_sp(A))
     => ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) ) ) )).

fof(fc2_coh_sp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k4_coh_sp(A))
      & v1_fraenkel(k4_coh_sp(A)) ) )).

fof(fc3_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k5_coh_sp(A)) )).

fof(fc4_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ( v1_relat_1(k2_mcart_1(B))
        & v1_funct_1(k2_mcart_1(B)) ) ) )).

fof(fc5_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k15_coh_sp(A)) )).

fof(fc6_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k16_coh_sp(A)) )).

fof(fc7_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k17_coh_sp(A)) )).

fof(fc8_coh_sp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k20_coh_sp(A))
      & v1_fraenkel(k20_coh_sp(A)) ) )).

fof(fc9_coh_sp,axiom,(
    ! [A] : ~ v1_xboole_0(k21_coh_sp(A)) )).

fof(fc10_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ( v1_relat_1(k2_mcart_1(B))
        & v1_funct_1(k2_mcart_1(B)) ) ) )).

fof(d1_coh_sp,axiom,(
    $true )).

fof(d2_coh_sp,axiom,(
    ! [A] :
      ( v1_coh_sp(A)
    <=> ! [B] :
          ( ( r1_tarski(B,A)
            & ! [C,D] :
                ( ( r2_hidden(C,B)
                  & r2_hidden(D,B) )
               => r2_hidden(k2_xboole_0(C,D),A) ) )
         => r2_hidden(k3_tarski(B),A) ) ) )).

fof(t1_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => r2_hidden(k1_xboole_0,A) ) )).

fof(t2_coh_sp,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v1_classes1(k1_zfmisc_1(A))
      & v1_coh_sp(k1_zfmisc_1(A)) ) )).

fof(t3_coh_sp,axiom,
    ( ~ v1_xboole_0(k1_tarski(k1_xboole_0))
    & v1_classes1(k1_tarski(k1_xboole_0))
    & v1_coh_sp(k1_tarski(k1_xboole_0)) )).

fof(t4_coh_sp,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( r2_hidden(A,k3_tarski(B))
       => r2_hidden(k1_tarski(A),B) ) ) )).

fof(d3_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ! [B] :
          ( ( v1_relat_2(B)
            & v3_relat_2(B)
            & v1_partfun1(B,k3_tarski(A),k3_tarski(A))
            & m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
         => ( B = k1_coh_sp(A)
          <=> ! [C,D] :
                ( r2_hidden(k4_tarski(C,D),B)
              <=> ? [E] :
                    ( r2_hidden(E,A)
                    & r2_hidden(C,E)
                    & r2_hidden(D,E) ) ) ) ) ) )).

fof(t5_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ! [B] :
          ( ( v1_relat_2(B)
            & v3_relat_2(B)
            & v1_partfun1(B,k3_tarski(A),k3_tarski(A))
            & m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
         => ( B = k1_coh_sp(A)
          <=> ! [C,D] :
                ( r2_hidden(k4_tarski(C,D),B)
              <=> r2_hidden(k2_tarski(C,D),A) ) ) ) ) )).

fof(t6_coh_sp,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( r2_hidden(A,B)
      <=> ! [C,D] :
            ( ( r2_hidden(C,A)
              & r2_hidden(D,A) )
           => r2_hidden(k2_tarski(C,D),B) ) ) ) )).

fof(t7_coh_sp,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( r2_hidden(A,B)
      <=> ! [C,D] :
            ( ( r2_hidden(C,A)
              & r2_hidden(D,A) )
           => r2_hidden(k4_tarski(C,D),k1_coh_sp(B)) ) ) ) )).

fof(t8_coh_sp,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( ! [C,D] :
            ( ( r2_hidden(C,A)
              & r2_hidden(D,A) )
           => r2_hidden(k2_tarski(C,D),B) )
       => r1_tarski(A,k3_tarski(B)) ) ) )).

fof(t9_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes1(B)
            & v1_coh_sp(B) )
         => ( k1_coh_sp(A) = k1_coh_sp(B)
           => A = B ) ) ) )).

fof(t10_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ( r2_hidden(k3_tarski(A),A)
       => A = k1_zfmisc_1(k3_tarski(A)) ) ) )).

fof(t11_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ( A = k1_zfmisc_1(k3_tarski(A))
       => k1_coh_sp(A) = k1_eqrel_1(k3_tarski(A)) ) ) )).

fof(d4_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & v1_classes1(C)
            & v1_coh_sp(C) )
         => ( C = k2_coh_sp(A,B)
          <=> ! [D] :
                ( r2_hidden(D,C)
              <=> ! [E,F] :
                    ( ( r2_hidden(E,D)
                      & r2_hidden(F,D) )
                   => r2_hidden(k4_tarski(E,F),B) ) ) ) ) ) )).

fof(t12_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k1_coh_sp(k2_coh_sp(A,B)) = B ) )).

fof(t13_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => k2_coh_sp(k3_tarski(A),k1_coh_sp(A)) = A ) )).

fof(t14_coh_sp,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ( r2_hidden(B,k2_coh_sp(A,C))
      <=> m1_toler_1(B,A,C) ) ) )).

fof(t15_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k2_coh_sp(A,B) = k3_toler_1(A,B) ) )).

fof(t16_coh_sp,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(D,k3_coh_sp(A))
     => ( r2_hidden(k2_tarski(B,C),D)
       => ( r2_hidden(B,k3_tarski(D))
          & r2_hidden(C,k3_tarski(D)) ) ) ) )).

fof(d6_coh_sp,axiom,(
    $true )).

fof(t17_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k4_coh_sp(B))
    <=> ? [C] :
          ( m1_subset_1(C,k3_coh_sp(B))
          & ? [D] :
              ( m1_subset_1(D,k3_coh_sp(B))
              & ( k3_tarski(D) = k1_xboole_0
               => k3_tarski(C) = k1_xboole_0 )
              & v1_funct_1(A)
              & v1_funct_2(A,k3_tarski(C),k3_tarski(D))
              & m2_relset_1(A,k3_tarski(C),k3_tarski(D)) ) ) ) )).

fof(t18_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ? [C] :
          ( m1_subset_1(C,k4_coh_sp(A))
          & ? [D] :
              ( m1_subset_1(D,k3_coh_sp(A))
              & ? [E] :
                  ( m1_subset_1(E,k3_coh_sp(A))
                  & B = k4_tarski(k4_tarski(D,E),C)
                  & ( k3_tarski(E) = k1_xboole_0
                   => k3_tarski(D) = k1_xboole_0 )
                  & v1_funct_1(C)
                  & v1_funct_2(C,k3_tarski(D),k3_tarski(E))
                  & m2_relset_1(C,k3_tarski(D),k3_tarski(E))
                  & ! [F,G] :
                      ( r2_hidden(k2_tarski(F,G),D)
                     => r2_hidden(k2_tarski(k1_funct_1(C,F),k1_funct_1(C,G)),E) ) ) ) ) ) )).

fof(t19_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k3_coh_sp(A))
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k3_tarski(B),k3_tarski(C))
                & m2_relset_1(D,k3_tarski(B),k3_tarski(C)) )
             => ( ! [E,F] :
                    ( r2_hidden(k2_tarski(E,F),B)
                   => r2_hidden(k2_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),C) )
               => ( ( k3_tarski(C) = k1_xboole_0
                    & k3_tarski(B) != k1_xboole_0 )
                  | r2_hidden(k4_tarski(k4_tarski(B,C),D),k5_coh_sp(A)) ) ) ) ) ) )).

fof(d9_coh_sp,axiom,(
    $true )).

fof(d10_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => k6_coh_sp(A,B) = k1_mcart_1(k1_mcart_1(B)) ) )).

fof(d11_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => k7_coh_sp(A,B) = k2_mcart_1(k1_mcart_1(B)) ) )).

fof(t20_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => B = k4_tarski(k4_tarski(k6_coh_sp(A,B),k7_coh_sp(A,B)),k2_mcart_1(B)) ) )).

fof(d12_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_coh_sp(A))
     => k8_coh_sp(A,B) = k4_tarski(k4_tarski(B,B),k6_partfun1(k3_tarski(B))) ) )).

fof(t21_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ( ~ ( k3_tarski(k7_coh_sp(A,B)) = k1_xboole_0
            & k3_tarski(k6_coh_sp(A,B)) != k1_xboole_0 )
        & v1_funct_1(k2_mcart_1(B))
        & v1_funct_2(k2_mcart_1(B),k3_tarski(k6_coh_sp(A,B)),k3_tarski(k7_coh_sp(A,B)))
        & m2_relset_1(k2_mcart_1(B),k3_tarski(k6_coh_sp(A,B)),k3_tarski(k7_coh_sp(A,B)))
        & ! [C,D] :
            ( r2_hidden(k2_tarski(C,D),k6_coh_sp(A,B))
           => r2_hidden(k2_tarski(k1_funct_1(k2_mcart_1(B),C),k1_funct_1(k2_mcart_1(B),D)),k7_coh_sp(A,B)) ) ) ) )).

fof(d13_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k5_coh_sp(A))
         => ( k7_coh_sp(A,B) = k6_coh_sp(A,C)
           => k9_coh_sp(A,B,C) = k4_tarski(k4_tarski(k6_coh_sp(A,B),k7_coh_sp(A,C)),k5_relat_1(k2_mcart_1(B),k2_mcart_1(C))) ) ) ) )).

fof(t22_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k5_coh_sp(A))
         => ( k6_coh_sp(A,B) = k7_coh_sp(A,C)
           => ( k2_mcart_1(k9_coh_sp(A,C,B)) = k5_relat_1(k2_mcart_1(C),k2_mcart_1(B))
              & k6_coh_sp(A,k9_coh_sp(A,C,B)) = k6_coh_sp(A,C)
              & k7_coh_sp(A,k9_coh_sp(A,C,B)) = k7_coh_sp(A,B) ) ) ) ) )).

fof(t23_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k5_coh_sp(A))
         => ! [D] :
              ( m1_subset_1(D,k5_coh_sp(A))
             => ( ( k6_coh_sp(A,B) = k7_coh_sp(A,C)
                  & k6_coh_sp(A,D) = k7_coh_sp(A,B) )
               => k9_coh_sp(A,k9_coh_sp(A,C,B),D) = k9_coh_sp(A,C,k9_coh_sp(A,B,D)) ) ) ) ) )).

fof(t24_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_coh_sp(A))
     => ( k2_mcart_1(k8_coh_sp(A,B)) = k6_partfun1(k3_tarski(B))
        & k6_coh_sp(A,k8_coh_sp(A,B)) = B
        & k7_coh_sp(A,k8_coh_sp(A,B)) = B ) ) )).

fof(t25_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => ( k9_coh_sp(A,k8_coh_sp(A,k6_coh_sp(A,B)),B) = B
        & k9_coh_sp(A,B,k8_coh_sp(A,k7_coh_sp(A,B))) = B ) ) )).

fof(d14_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_coh_sp(A),k3_coh_sp(A))
        & m2_relset_1(B,k5_coh_sp(A),k3_coh_sp(A)) )
     => ( B = k10_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k5_coh_sp(A))
           => k8_funct_2(k5_coh_sp(A),k3_coh_sp(A),B,C) = k6_coh_sp(A,C) ) ) ) )).

fof(d15_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_coh_sp(A),k3_coh_sp(A))
        & m2_relset_1(B,k5_coh_sp(A),k3_coh_sp(A)) )
     => ( B = k11_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k5_coh_sp(A))
           => k8_funct_2(k5_coh_sp(A),k3_coh_sp(A),B,C) = k7_coh_sp(A,C) ) ) ) )).

fof(d16_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A)) )
     => ( B = k12_coh_sp(A)
      <=> ( ! [C] :
              ( m1_subset_1(C,k5_coh_sp(A))
             => ! [D] :
                  ( m1_subset_1(D,k5_coh_sp(A))
                 => ( r2_hidden(k4_tarski(C,D),k4_relset_1(k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A),B))
                  <=> k6_coh_sp(A,C) = k7_coh_sp(A,D) ) ) )
          & ! [C] :
              ( m1_subset_1(C,k5_coh_sp(A))
             => ! [D] :
                  ( m1_subset_1(D,k5_coh_sp(A))
                 => ( k6_coh_sp(A,C) = k7_coh_sp(A,D)
                   => k1_funct_1(B,k4_tarski(C,D)) = k9_coh_sp(A,D,C) ) ) ) ) ) ) )).

fof(d17_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k3_coh_sp(A),k5_coh_sp(A))
        & m2_relset_1(B,k3_coh_sp(A),k5_coh_sp(A)) )
     => ( B = k13_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k3_coh_sp(A))
           => k8_funct_2(k3_coh_sp(A),k5_coh_sp(A),B,C) = k8_coh_sp(A,C) ) ) ) )).

fof(t26_coh_sp,axiom,(
    ! [A] :
      ( v2_cat_1(g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A)))
      & l1_cat_1(g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A))) ) )).

fof(d18_coh_sp,axiom,(
    ! [A] : k14_coh_sp(A) = g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A)) )).

fof(d19_coh_sp,axiom,(
    ! [A,B] :
      ( B = k15_coh_sp(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) ) ) ) )).

fof(t27_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k16_coh_sp(B))
    <=> ? [C] :
          ( r1_tarski(C,B)
          & v1_relat_2(A)
          & v3_relat_2(A)
          & v1_partfun1(A,C,C)
          & m2_relset_1(A,C,C) ) ) )).

fof(t28_coh_sp,axiom,(
    ! [A] : r2_hidden(k1_eqrel_1(A),k15_coh_sp(A)) )).

fof(t29_coh_sp,axiom,(
    $true )).

fof(t30_coh_sp,axiom,(
    ! [A] : r2_hidden(k1_xboole_0,k16_coh_sp(A)) )).

fof(t31_coh_sp,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r2_hidden(k1_eqrel_1(A),k16_coh_sp(B)) ) )).

fof(t32_coh_sp,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r2_hidden(k6_partfun1(A),k16_coh_sp(B)) ) )).

fof(t33_coh_sp,axiom,(
    ! [A] : r2_hidden(k1_eqrel_1(A),k16_coh_sp(A)) )).

fof(t34_coh_sp,axiom,(
    ! [A] : r2_hidden(k6_partfun1(A),k16_coh_sp(A)) )).

fof(t35_coh_sp,axiom,(
    ! [A] : r2_hidden(k4_tarski(k1_xboole_0,k1_xboole_0),k17_coh_sp(A)) )).

fof(t36_coh_sp,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r2_hidden(k4_tarski(k6_partfun1(A),A),k17_coh_sp(B)) ) )).

fof(t37_coh_sp,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r2_hidden(k4_tarski(k1_eqrel_1(A),A),k17_coh_sp(B)) ) )).

fof(t38_coh_sp,axiom,(
    ! [A] : r2_hidden(k4_tarski(k6_partfun1(A),A),k17_coh_sp(A)) )).

fof(t39_coh_sp,axiom,(
    ! [A] : r2_hidden(k4_tarski(k1_eqrel_1(A),A),k17_coh_sp(A)) )).

fof(t40_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k20_coh_sp(B))
    <=> ? [C] :
          ( m1_subset_1(C,k17_coh_sp(B))
          & ? [D] :
              ( m1_subset_1(D,k17_coh_sp(B))
              & ( k18_coh_sp(B,D) = k1_xboole_0
               => k18_coh_sp(B,C) = k1_xboole_0 )
              & v1_funct_1(A)
              & v1_funct_2(A,k18_coh_sp(B,C),k18_coh_sp(B,D))
              & m2_relset_1(A,k18_coh_sp(B,C),k18_coh_sp(B,D)) ) ) ) )).

fof(t41_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ? [C] :
          ( m1_subset_1(C,k20_coh_sp(A))
          & ? [D] :
              ( m1_subset_1(D,k17_coh_sp(A))
              & ? [E] :
                  ( m1_subset_1(E,k17_coh_sp(A))
                  & B = k4_tarski(k4_tarski(D,E),C)
                  & ( k18_coh_sp(A,E) = k1_xboole_0
                   => k18_coh_sp(A,D) = k1_xboole_0 )
                  & v1_funct_1(C)
                  & v1_funct_2(C,k18_coh_sp(A,D),k18_coh_sp(A,E))
                  & m2_relset_1(C,k18_coh_sp(A,D),k18_coh_sp(A,E))
                  & ! [F,G] :
                      ( r2_hidden(k4_tarski(F,G),k19_coh_sp(A,D))
                     => r2_hidden(k4_tarski(k1_funct_1(C,F),k1_funct_1(C,G)),k19_coh_sp(A,E)) ) ) ) ) ) )).

fof(t42_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k17_coh_sp(A))
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k18_coh_sp(A,B),k18_coh_sp(A,C))
                & m2_relset_1(D,k18_coh_sp(A,B),k18_coh_sp(A,C)) )
             => ( ! [E,F] :
                    ( r2_hidden(k4_tarski(E,F),k19_coh_sp(A,B))
                   => r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),k19_coh_sp(A,C)) )
               => ( ( k18_coh_sp(A,C) = k1_xboole_0
                    & k18_coh_sp(A,B) != k1_xboole_0 )
                  | r2_hidden(k4_tarski(k4_tarski(B,C),D),k21_coh_sp(A)) ) ) ) ) ) )).

fof(d24_coh_sp,axiom,(
    $true )).

fof(d25_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => k22_coh_sp(A,B) = k1_mcart_1(k1_mcart_1(B)) ) )).

fof(d26_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => k23_coh_sp(A,B) = k2_mcart_1(k1_mcart_1(B)) ) )).

fof(t43_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => B = k4_tarski(k4_tarski(k22_coh_sp(A,B),k23_coh_sp(A,B)),k2_mcart_1(B)) ) )).

fof(d27_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => k24_coh_sp(A,B) = k4_tarski(k4_tarski(B,B),k6_partfun1(k18_coh_sp(A,B))) ) )).

fof(t44_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ( ~ ( k18_coh_sp(A,k23_coh_sp(A,B)) = k1_xboole_0
            & k18_coh_sp(A,k22_coh_sp(A,B)) != k1_xboole_0 )
        & v1_funct_1(k2_mcart_1(B))
        & v1_funct_2(k2_mcart_1(B),k18_coh_sp(A,k22_coh_sp(A,B)),k18_coh_sp(A,k23_coh_sp(A,B)))
        & m2_relset_1(k2_mcart_1(B),k18_coh_sp(A,k22_coh_sp(A,B)),k18_coh_sp(A,k23_coh_sp(A,B)))
        & ! [C,D] :
            ( r2_hidden(k4_tarski(C,D),k19_coh_sp(A,k22_coh_sp(A,B)))
           => r2_hidden(k4_tarski(k1_funct_1(k2_mcart_1(B),C),k1_funct_1(k2_mcart_1(B),D)),k19_coh_sp(A,k23_coh_sp(A,B))) ) ) ) )).

fof(d28_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k21_coh_sp(A))
         => ( k23_coh_sp(A,B) = k22_coh_sp(A,C)
           => k25_coh_sp(A,B,C) = k4_tarski(k4_tarski(k22_coh_sp(A,B),k23_coh_sp(A,C)),k5_relat_1(k2_mcart_1(B),k2_mcart_1(C))) ) ) ) )).

fof(t45_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k21_coh_sp(A))
         => ( k22_coh_sp(A,B) = k23_coh_sp(A,C)
           => ( k2_mcart_1(k25_coh_sp(A,C,B)) = k5_relat_1(k2_mcart_1(C),k2_mcart_1(B))
              & k22_coh_sp(A,k25_coh_sp(A,C,B)) = k22_coh_sp(A,C)
              & k23_coh_sp(A,k25_coh_sp(A,C,B)) = k23_coh_sp(A,B) ) ) ) ) )).

fof(t46_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ! [C] :
          ( m1_subset_1(C,k21_coh_sp(A))
         => ! [D] :
              ( m1_subset_1(D,k21_coh_sp(A))
             => ( ( k22_coh_sp(A,B) = k23_coh_sp(A,C)
                  & k22_coh_sp(A,D) = k23_coh_sp(A,B) )
               => k25_coh_sp(A,k25_coh_sp(A,C,B),D) = k25_coh_sp(A,C,k25_coh_sp(A,B,D)) ) ) ) ) )).

fof(t47_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => ( k2_mcart_1(k24_coh_sp(A,B)) = k6_partfun1(k18_coh_sp(A,B))
        & k22_coh_sp(A,k24_coh_sp(A,B)) = B
        & k23_coh_sp(A,k24_coh_sp(A,B)) = B ) ) )).

fof(t48_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => ( k25_coh_sp(A,k24_coh_sp(A,k22_coh_sp(A,B)),B) = B
        & k25_coh_sp(A,B,k24_coh_sp(A,k23_coh_sp(A,B))) = B ) ) )).

fof(d29_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k21_coh_sp(A),k17_coh_sp(A))
        & m2_relset_1(B,k21_coh_sp(A),k17_coh_sp(A)) )
     => ( B = k26_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k21_coh_sp(A))
           => k8_funct_2(k21_coh_sp(A),k17_coh_sp(A),B,C) = k22_coh_sp(A,C) ) ) ) )).

fof(d30_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k21_coh_sp(A),k17_coh_sp(A))
        & m2_relset_1(B,k21_coh_sp(A),k17_coh_sp(A)) )
     => ( B = k27_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k21_coh_sp(A))
           => k8_funct_2(k21_coh_sp(A),k17_coh_sp(A),B,C) = k23_coh_sp(A,C) ) ) ) )).

fof(d31_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A)) )
     => ( B = k28_coh_sp(A)
      <=> ( ! [C] :
              ( m1_subset_1(C,k21_coh_sp(A))
             => ! [D] :
                  ( m1_subset_1(D,k21_coh_sp(A))
                 => ( r2_hidden(k4_tarski(C,D),k4_relset_1(k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A),B))
                  <=> k22_coh_sp(A,C) = k23_coh_sp(A,D) ) ) )
          & ! [C] :
              ( m1_subset_1(C,k21_coh_sp(A))
             => ! [D] :
                  ( m1_subset_1(D,k21_coh_sp(A))
                 => ( k22_coh_sp(A,C) = k23_coh_sp(A,D)
                   => k1_funct_1(B,k4_tarski(C,D)) = k25_coh_sp(A,D,C) ) ) ) ) ) ) )).

fof(d32_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k17_coh_sp(A),k21_coh_sp(A))
        & m2_relset_1(B,k17_coh_sp(A),k21_coh_sp(A)) )
     => ( B = k29_coh_sp(A)
      <=> ! [C] :
            ( m1_subset_1(C,k17_coh_sp(A))
           => k8_funct_2(k17_coh_sp(A),k21_coh_sp(A),B,C) = k24_coh_sp(A,C) ) ) ) )).

fof(t49_coh_sp,axiom,(
    ! [A] :
      ( v2_cat_1(g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A)))
      & l1_cat_1(g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A))) ) )).

fof(d33_coh_sp,axiom,(
    ! [A] : k30_coh_sp(A) = g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A)) )).

fof(dt_k1_coh_sp,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ( v1_relat_2(k1_coh_sp(A))
        & v3_relat_2(k1_coh_sp(A))
        & v1_partfun1(k1_coh_sp(A),k3_tarski(A),k3_tarski(A))
        & m2_relset_1(k1_coh_sp(A),k3_tarski(A),k3_tarski(A)) ) ) )).

fof(dt_k2_coh_sp,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ( ~ v1_xboole_0(k2_coh_sp(A,B))
        & v1_classes1(k2_coh_sp(A,B))
        & v1_coh_sp(k2_coh_sp(A,B)) ) ) )).

fof(dt_k3_coh_sp,axiom,(
    $true )).

fof(dt_k4_coh_sp,axiom,(
    $true )).

fof(dt_k5_coh_sp,axiom,(
    $true )).

fof(dt_k6_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => m1_subset_1(k6_coh_sp(A,B),k3_coh_sp(A)) ) )).

fof(dt_k7_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_coh_sp(A))
     => m1_subset_1(k7_coh_sp(A,B),k3_coh_sp(A)) ) )).

fof(dt_k8_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_coh_sp(A))
     => m1_subset_1(k8_coh_sp(A,B),k5_coh_sp(A)) ) )).

fof(dt_k9_coh_sp,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k5_coh_sp(A))
        & m1_subset_1(C,k5_coh_sp(A)) )
     => m1_subset_1(k9_coh_sp(A,B,C),k5_coh_sp(A)) ) )).

fof(dt_k10_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k10_coh_sp(A))
      & v1_funct_2(k10_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A))
      & m2_relset_1(k10_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A)) ) )).

fof(dt_k11_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k11_coh_sp(A))
      & v1_funct_2(k11_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A))
      & m2_relset_1(k11_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A)) ) )).

fof(dt_k12_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k12_coh_sp(A))
      & m2_relset_1(k12_coh_sp(A),k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A)) ) )).

fof(dt_k13_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k13_coh_sp(A))
      & v1_funct_2(k13_coh_sp(A),k3_coh_sp(A),k5_coh_sp(A))
      & m2_relset_1(k13_coh_sp(A),k3_coh_sp(A),k5_coh_sp(A)) ) )).

fof(dt_k14_coh_sp,axiom,(
    ! [A] :
      ( v2_cat_1(k14_coh_sp(A))
      & l1_cat_1(k14_coh_sp(A)) ) )).

fof(dt_k15_coh_sp,axiom,(
    $true )).

fof(dt_k16_coh_sp,axiom,(
    $true )).

fof(dt_k17_coh_sp,axiom,(
    $true )).

fof(dt_k18_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => m1_subset_1(k18_coh_sp(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k18_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => k18_coh_sp(A,B) = k2_mcart_1(B) ) )).

fof(dt_k19_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => ( v1_relat_2(k19_coh_sp(A,B))
        & v3_relat_2(k19_coh_sp(A,B))
        & v1_partfun1(k19_coh_sp(A,B),k2_mcart_1(B),k2_mcart_1(B))
        & m2_relset_1(k19_coh_sp(A,B),k2_mcart_1(B),k2_mcart_1(B)) ) ) )).

fof(redefinition_k19_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => k19_coh_sp(A,B) = k1_mcart_1(B) ) )).

fof(dt_k20_coh_sp,axiom,(
    $true )).

fof(dt_k21_coh_sp,axiom,(
    $true )).

fof(dt_k22_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => m1_subset_1(k22_coh_sp(A,B),k17_coh_sp(A)) ) )).

fof(dt_k23_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k21_coh_sp(A))
     => m1_subset_1(k23_coh_sp(A,B),k17_coh_sp(A)) ) )).

fof(dt_k24_coh_sp,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k17_coh_sp(A))
     => m1_subset_1(k24_coh_sp(A,B),k21_coh_sp(A)) ) )).

fof(dt_k25_coh_sp,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k21_coh_sp(A))
        & m1_subset_1(C,k21_coh_sp(A)) )
     => m1_subset_1(k25_coh_sp(A,B,C),k21_coh_sp(A)) ) )).

fof(dt_k26_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k26_coh_sp(A))
      & v1_funct_2(k26_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A))
      & m2_relset_1(k26_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A)) ) )).

fof(dt_k27_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k27_coh_sp(A))
      & v1_funct_2(k27_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A))
      & m2_relset_1(k27_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A)) ) )).

fof(dt_k28_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k28_coh_sp(A))
      & m2_relset_1(k28_coh_sp(A),k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A)) ) )).

fof(dt_k29_coh_sp,axiom,(
    ! [A] :
      ( v1_funct_1(k29_coh_sp(A))
      & v1_funct_2(k29_coh_sp(A),k17_coh_sp(A),k21_coh_sp(A))
      & m2_relset_1(k29_coh_sp(A),k17_coh_sp(A),k21_coh_sp(A)) ) )).

fof(dt_k30_coh_sp,axiom,(
    ! [A] :
      ( v2_cat_1(k30_coh_sp(A))
      & l1_cat_1(k30_coh_sp(A)) ) )).

fof(d5_coh_sp,axiom,(
    ! [A] : k3_coh_sp(A) = a_1_0_coh_sp(A) )).

fof(d7_coh_sp,axiom,(
    ! [A] : k4_coh_sp(A) = k3_tarski(a_1_1_coh_sp(A)) )).

fof(d8_coh_sp,axiom,(
    ! [A] : k5_coh_sp(A) = a_1_2_coh_sp(A) )).

fof(d20_coh_sp,axiom,(
    ! [A] : k16_coh_sp(A) = k3_tarski(a_1_3_coh_sp(A)) )).

fof(d21_coh_sp,axiom,(
    ! [A] : k17_coh_sp(A) = a_1_4_coh_sp(A) )).

fof(d22_coh_sp,axiom,(
    ! [A] : k20_coh_sp(A) = k3_tarski(a_1_5_coh_sp(A)) )).

fof(d23_coh_sp,axiom,(
    ! [A] : k21_coh_sp(A) = a_1_6_coh_sp(A) )).

fof(fraenkel_a_1_0_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_0_coh_sp(B))
    <=> ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
          & A = C
          & ~ v1_xboole_0(C)
          & v1_classes1(C)
          & v1_coh_sp(C) ) ) )).

fof(fraenkel_a_1_1_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_1_coh_sp(B))
    <=> ? [C,D] :
          ( m1_subset_1(C,k3_coh_sp(B))
          & m1_subset_1(D,k3_coh_sp(B))
          & A = k1_funct_2(k3_tarski(C),k3_tarski(D)) ) ) )).

fof(fraenkel_a_1_2_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_2_coh_sp(B))
    <=> ? [C,D,E] :
          ( m1_subset_1(C,k3_coh_sp(B))
          & m1_subset_1(D,k3_coh_sp(B))
          & m1_subset_1(E,k4_coh_sp(B))
          & A = k4_tarski(k4_tarski(C,D),E)
          & ( k3_tarski(D) = k1_xboole_0
           => k3_tarski(C) = k1_xboole_0 )
          & v1_funct_1(E)
          & v1_funct_2(E,k3_tarski(C),k3_tarski(D))
          & m2_relset_1(E,k3_tarski(C),k3_tarski(D))
          & ! [F,G] :
              ( r2_hidden(k2_tarski(F,G),C)
             => r2_hidden(k2_tarski(k1_funct_1(E,F),k1_funct_1(E,G)),D) ) ) ) )).

fof(fraenkel_a_1_3_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_3_coh_sp(B))
    <=> ? [C] :
          ( m1_subset_1(C,k1_zfmisc_1(B))
          & A = k15_coh_sp(C) ) ) )).

fof(fraenkel_a_1_4_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_4_coh_sp(B))
    <=> ? [C,D] :
          ( m1_subset_1(C,k16_coh_sp(B))
          & m1_subset_1(D,k1_zfmisc_1(B))
          & A = k4_tarski(C,D)
          & v1_relat_2(C)
          & v3_relat_2(C)
          & v1_partfun1(C,D,D)
          & m2_relset_1(C,D,D) ) ) )).

fof(fraenkel_a_1_5_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_5_coh_sp(B))
    <=> ? [C,D] :
          ( m1_subset_1(C,k17_coh_sp(B))
          & m1_subset_1(D,k17_coh_sp(B))
          & A = k1_funct_2(k18_coh_sp(B,C),k18_coh_sp(B,D)) ) ) )).

fof(fraenkel_a_1_6_coh_sp,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_6_coh_sp(B))
    <=> ? [C,D,E] :
          ( m1_subset_1(C,k17_coh_sp(B))
          & m1_subset_1(D,k17_coh_sp(B))
          & m1_subset_1(E,k20_coh_sp(B))
          & A = k4_tarski(k4_tarski(C,D),E)
          & ( k18_coh_sp(B,D) = k1_xboole_0
           => k18_coh_sp(B,C) = k1_xboole_0 )
          & v1_funct_1(E)
          & v1_funct_2(E,k18_coh_sp(B,C),k18_coh_sp(B,D))
          & m2_relset_1(E,k18_coh_sp(B,C),k18_coh_sp(B,D))
          & ! [F,G] :
              ( r2_hidden(k4_tarski(F,G),k19_coh_sp(B,C))
             => r2_hidden(k4_tarski(k1_funct_1(E,F),k1_funct_1(E,G)),k19_coh_sp(B,D)) ) ) ) )).
%------------------------------------------------------------------------------