SET007 Axioms: SET007+841.ax


%------------------------------------------------------------------------------
% File     : SET007+841 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Hall Marriage Theorem
% 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    : hallmar1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   57 (   1 unit)
%            Number of atoms       :  338 (  33 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  319 (  38 ~  ;   6  |;  85  &)
%                                         (  11 <=>; 179 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-4 arity)
%            Number of functors    :   21 (   6 constant; 0-4 arity)
%            Number of variables   :  209 (   0 singleton; 199 !;  10 ?)
%            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_hallmar1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_finseq_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_relat_1(B)
          & v2_relat_1(B)
          & v1_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B) ) ) )).

fof(fc1_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(A)
        & m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => v1_finset_1(k1_funct_1(B,C)) ) )).

fof(fc2_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(A)
        & m1_finseq_1(B,k1_zfmisc_1(A)) )
     => v1_finset_1(k1_hallmar1(A,B,C)) ) )).

fof(cc1_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & ~ v1_xboole_0(B)
        & m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => ! [D] :
          ( m3_hallmar1(D,A,B,C)
         => ( ~ v1_xboole_0(D)
            & v1_relat_1(D) ) ) ) )).

fof(cc2_hallmar1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & ~ v1_xboole_0(B)
        & m1_finseq_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m4_hallmar1(C,A,B)
         => ( ~ v1_xboole_0(C)
            & v1_relat_1(C) ) ) ) )).

fof(t1_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => k1_nat_1(k4_card_1(k2_xboole_0(A,B)),k4_card_1(k3_xboole_0(A,B))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) )).

fof(t2_hallmar1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_relat_1(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(C,k4_finseq_1(B))
                  & k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) )).

fof(d1_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C,D] :
          ( D = k1_hallmar1(A,B,C)
        <=> ! [E] :
              ( r2_hidden(E,D)
            <=> ? [F] :
                  ( r2_hidden(F,C)
                  & r2_hidden(F,k4_finseq_1(B))
                  & r2_hidden(E,k1_funct_1(B,F)) ) ) ) ) )).

fof(t3_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] : r1_tarski(k1_hallmar1(A,B,C),A) ) )).

fof(t4_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C,D] :
              ( r1_tarski(C,D)
             => r1_tarski(k1_hallmar1(A,B,C),k1_hallmar1(A,B,D)) ) ) ) )).

fof(t5_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(B))
               => k1_hallmar1(A,B,k1_tarski(C)) = k1_funct_1(B,C) ) ) ) ) )).

fof(t6_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(C,k4_finseq_1(B))
                      & r2_hidden(D,k4_finseq_1(B)) )
                   => k1_hallmar1(A,B,k2_tarski(C,D)) = k2_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) )).

fof(t7_hallmar1,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_finseq_1(C,k1_zfmisc_1(B))
         => ! [D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => ( ( r2_hidden(D,A)
                  & r2_hidden(D,k4_finseq_1(C)) )
               => r1_tarski(k1_funct_1(C,D),k1_hallmar1(B,C,A)) ) ) ) ) )).

fof(t8_hallmar1,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( m2_finseq_1(D,k1_zfmisc_1(B))
             => ( ( r2_hidden(C,A)
                  & r2_hidden(C,k4_finseq_1(D)) )
               => k1_hallmar1(B,D,A) = k2_xboole_0(k1_hallmar1(B,D,k4_xboole_0(A,k1_tarski(C))),k1_funct_1(D,C)) ) ) ) ) )).

fof(t9_hallmar1,axiom,(
    ! [A,B,C] :
      ( v1_finset_1(C)
     => ! [D] :
          ( m2_subset_1(D,k1_numbers,k5_numbers)
         => ! [E] :
              ( m2_finseq_1(E,k1_zfmisc_1(C))
             => ( r2_hidden(D,k4_finseq_1(E))
               => k1_hallmar1(C,E,k2_xboole_0(k2_xboole_0(k1_tarski(D),A),B)) = k2_xboole_0(k1_funct_1(E,D),k1_hallmar1(C,E,k2_xboole_0(A,B))) ) ) ) ) )).

fof(t10_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D,E] :
                  ( ( r2_hidden(D,k1_funct_1(B,C))
                    & r2_hidden(E,k1_funct_1(B,C)) )
                 => ( D = E
                    | k2_xboole_0(k4_xboole_0(k1_funct_1(B,C),k1_tarski(D)),k4_xboole_0(k1_funct_1(B,C),k1_tarski(E))) = k1_funct_1(B,C) ) ) ) ) ) )).

fof(d2_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D,E] :
                  ( m2_finseq_1(E,k1_zfmisc_1(A))
                 => ( E = k2_hallmar1(A,B,C,D)
                  <=> ( k4_finseq_1(E) = k4_finseq_1(B)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( r2_hidden(F,k4_finseq_1(E))
                           => ( ( C = F
                               => k1_funct_1(E,F) = k4_xboole_0(k1_funct_1(B,F),k1_tarski(D)) )
                              & ( C != F
                               => k1_funct_1(E,F) = k1_funct_1(B,F) ) ) ) ) ) ) ) ) ) ) )).

fof(t11_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( r2_hidden(C,k4_finseq_1(B))
                    & r2_hidden(D,k1_funct_1(B,C)) )
                 => k4_card_1(k1_funct_1(k2_hallmar1(A,B,C,D),C)) = k5_real_1(k4_card_1(k1_funct_1(B,C)),np__1) ) ) ) ) )).

fof(t12_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D,E] : k1_hallmar1(A,k2_hallmar1(A,B,C,D),k4_xboole_0(E,k1_tarski(C))) = k1_hallmar1(A,B,k4_xboole_0(E,k1_tarski(C))) ) ) ) )).

fof(t13_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D,E] :
                  ( ~ r2_hidden(C,E)
                 => k1_hallmar1(A,B,E) = k1_hallmar1(A,k2_hallmar1(A,B,C,D),E) ) ) ) ) )).

fof(t14_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D,E] :
                  ( ( r2_hidden(C,k4_finseq_1(k2_hallmar1(A,B,C,D)))
                    & r1_tarski(E,k4_finseq_1(k2_hallmar1(A,B,C,D)))
                    & r2_hidden(C,E) )
                 => k1_hallmar1(A,k2_hallmar1(A,B,C,D),E) = k2_xboole_0(k1_hallmar1(A,B,k4_xboole_0(E,k1_tarski(C))),k4_xboole_0(k1_funct_1(B,C),k1_tarski(D))) ) ) ) ) )).

fof(d3_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( r1_hallmar1(A,B,C)
            <=> ? [D] :
                  ( m2_finseq_1(D,A)
                  & D = C
                  & k4_finseq_1(B) = k4_finseq_1(D)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k4_finseq_1(D))
                       => r2_hidden(k1_funct_1(D,E),k1_funct_1(B,E)) ) )
                  & v2_funct_1(D) ) ) ) ) )).

fof(d4_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ( v1_hallmar1(B,A)
          <=> ! [C] :
                ( v1_finset_1(C)
               => ( r1_tarski(C,k4_finseq_1(B))
                 => r1_xreal_0(k4_card_1(C),k4_card_1(k1_hallmar1(A,B,C))) ) ) ) ) ) )).

fof(t15_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ( v1_hallmar1(B,A)
           => v2_relat_1(B) ) ) ) )).

fof(t16_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(C,k4_finseq_1(B))
                  & v1_hallmar1(B,A) )
               => r1_xreal_0(np__1,k4_card_1(k1_funct_1(B,C))) ) ) ) ) )).

fof(t17_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => k4_card_1(k1_funct_1(B,C)) = np__1 ) )
              & v1_hallmar1(B,A)
              & ! [C] : ~ r1_hallmar1(A,B,C) ) ) ) )).

fof(t18_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ( ? [C] : r1_hallmar1(A,B,C)
           => v1_hallmar1(B,A) ) ) ) )).

fof(d5_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( m2_finseq_1(D,k1_zfmisc_1(A))
             => ( m1_hallmar1(D,A,B,C)
              <=> ( k4_finseq_1(D) = k4_finseq_1(B)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k4_finseq_1(B))
                       => ( E = C
                          | k1_funct_1(B,E) = k1_funct_1(D,E) ) ) )
                  & r1_tarski(k1_funct_1(D,C),k1_funct_1(B,C)) ) ) ) ) ) )).

fof(d6_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_finseq_1(C,k1_zfmisc_1(A))
         => ( m2_hallmar1(C,A,B)
          <=> ( k4_finseq_1(C) = k4_finseq_1(B)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_hidden(D,k4_finseq_1(B))
                   => r1_tarski(k1_funct_1(C,D),k1_funct_1(B,D)) ) ) ) ) ) ) )).

fof(d7_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r2_hidden(C,k4_finseq_1(B))
           => ( k1_funct_1(B,C) = k1_xboole_0
              | ! [D] :
                  ( m2_hallmar1(D,A,B)
                 => ( m3_hallmar1(D,A,B,C)
                  <=> k1_card_1(k1_funct_1(D,C)) = np__1 ) ) ) ) ) ) )).

fof(t19_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_hallmar1(D,A,B,C)
                 => m2_hallmar1(D,A,B) ) ) ) ) )).

fof(t20_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( r2_hidden(C,k4_finseq_1(B))
                    & r2_hidden(D,k1_funct_1(B,C)) )
                 => m1_hallmar1(k2_hallmar1(A,B,C,D),A,B,C) ) ) ) ) )).

fof(t21_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( r2_hidden(C,k4_finseq_1(B))
                    & r2_hidden(D,k1_funct_1(B,C)) )
                 => m2_hallmar1(k2_hallmar1(A,B,C,D),A,B) ) ) ) ) )).

fof(t22_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_hallmar1(C,A,B)
             => ! [D] :
                  ( m2_hallmar1(D,A,C)
                 => m2_hallmar1(D,A,B) ) ) ) ) )).

fof(t23_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m3_hallmar1(D,A,B,C)
                 => ~ ( r2_hidden(C,k4_finseq_1(B))
                      & k1_funct_1(D,C) = k1_xboole_0 ) ) ) ) ) )).

fof(t24_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m3_hallmar1(E,A,B,C)
                     => ! [F] :
                          ( m3_hallmar1(F,A,E,D)
                         => ( ( r2_hidden(C,k4_finseq_1(B))
                              & r2_hidden(D,k4_finseq_1(B)) )
                           => ( k1_funct_1(F,C) = k1_xboole_0
                              | k1_funct_1(E,D) = k1_xboole_0
                              | ( m3_hallmar1(F,A,B,D)
                                & m3_hallmar1(F,A,B,C) ) ) ) ) ) ) ) ) ) )).

fof(t25_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => m1_hallmar1(B,A,B,C) ) ) )).

fof(t26_hallmar1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_zfmisc_1(A))
     => m2_hallmar1(B,A,B) ) )).

fof(d8_hallmar1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ( v2_relat_1(B)
           => ! [C] :
                ( m2_hallmar1(C,A,B)
               => ( m4_hallmar1(C,A,B)
                <=> ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => k1_card_1(k1_funct_1(C,D)) = np__1 ) ) ) ) ) ) ) )).

fof(t27_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_relat_1(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( m4_hallmar1(C,A,B)
              <=> ( k1_relat_1(C) = k4_finseq_1(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => m3_hallmar1(C,A,B,D) ) ) ) ) ) ) ) )).

fof(t28_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ! [C,D] :
              ( m2_hallmar1(D,A,B)
             => ( r1_hallmar1(A,D,C)
               => r1_hallmar1(A,B,C) ) ) ) ) )).

fof(t29_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ( v1_hallmar1(B,A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r1_xreal_0(np__2,k4_card_1(k1_funct_1(B,C)))
                    & ! [D] :
                        ~ ( r2_hidden(D,k1_funct_1(B,C))
                          & v1_hallmar1(k2_hallmar1(A,B,C,D),A) ) ) ) ) ) ) )).

fof(t30_hallmar1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(C,k4_finseq_1(B))
                  & v1_hallmar1(B,A)
                  & ! [D] :
                      ( m3_hallmar1(D,A,B,C)
                     => ~ v1_hallmar1(D,A) ) ) ) ) ) )).

fof(t31_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ~ ( v1_hallmar1(B,A)
              & ! [C] :
                  ( m4_hallmar1(C,A,B)
                 => ~ v1_hallmar1(C,A) ) ) ) ) )).

fof(t32_hallmar1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,k1_zfmisc_1(A)) )
         => ( ? [C] : r1_hallmar1(A,B,C)
          <=> v1_hallmar1(B,A) ) ) ) )).

fof(s1_hallmar1,axiom,
    ( ( p1_s1_hallmar1(f1_s1_hallmar1)
      & r1_xreal_0(np__2,f1_s1_hallmar1)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & p1_s1_hallmar1(k1_nat_1(A,np__1)) )
           => ( r1_xreal_0(f1_s1_hallmar1,A)
              | p1_s1_hallmar1(A) ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(np__1,A)
            & r1_xreal_0(A,f1_s1_hallmar1) )
         => p1_s1_hallmar1(A) ) ) )).

fof(s2_hallmar1,axiom,
    ( ( ? [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
          & ~ r1_xreal_0(A,np__1)
          & p1_s2_hallmar1(A) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & p1_s2_hallmar1(k1_nat_1(A,np__1)) )
           => p1_s2_hallmar1(A) ) ) )
   => p1_s2_hallmar1(np__1) )).

fof(dt_m1_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => ! [D] :
          ( m1_hallmar1(D,A,B,C)
         => m2_finseq_1(D,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => ? [D] : m1_hallmar1(D,A,B,C) ) )).

fof(dt_m2_hallmar1,axiom,(
    ! [A,B] :
      ( m1_finseq_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m2_hallmar1(C,A,B)
         => m2_finseq_1(C,k1_zfmisc_1(A)) ) ) )).

fof(existence_m2_hallmar1,axiom,(
    ! [A,B] :
      ( m1_finseq_1(B,k1_zfmisc_1(A))
     => ? [C] : m2_hallmar1(C,A,B) ) )).

fof(dt_m3_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => ! [D] :
          ( m3_hallmar1(D,A,B,C)
         => m2_hallmar1(D,A,B) ) ) )).

fof(existence_m3_hallmar1,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => ? [D] : m3_hallmar1(D,A,B,C) ) )).

fof(dt_m4_hallmar1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m4_hallmar1(C,A,B)
         => m2_hallmar1(C,A,B) ) ) )).

fof(existence_m4_hallmar1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m4_hallmar1(C,A,B) ) )).

fof(dt_k1_hallmar1,axiom,(
    $true )).

fof(dt_k2_hallmar1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_finset_1(A)
        & m1_finseq_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k2_hallmar1(A,B,C,D),k1_zfmisc_1(A)) ) )).
%------------------------------------------------------------------------------