SET007 Axioms: SET007+114.ax


%------------------------------------------------------------------------------
% File     : SET007+114 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Schemes of Existence of Some Types of Functions
% 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    : scheme1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   26 (   0 unit)
%            Number of atoms       :  472 (  81 equality)
%            Maximal formula depth :   16 (  12 average)
%            Number of connectives :  555 ( 109 ~  ;   5  |; 267  &)
%                                         (  13 <=>; 161 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   56 (   0 propositional; 1-3 arity)
%            Number of functors    :  117 (  48 constant; 0-4 arity)
%            Number of variables   :  124 (   0 singleton; 101 !;  23 ?)
%            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(t1_scheme1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( A != k2_nat_1(np__2,B)
              & A != k1_nat_1(k2_nat_1(np__2,B),np__1) ) ) ) )).

fof(t2_scheme1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( A != k2_nat_1(np__3,B)
              & A != k1_nat_1(k2_nat_1(np__3,B),np__1)
              & A != k1_nat_1(k2_nat_1(np__3,B),np__2) ) ) ) )).

fof(t3_scheme1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( A != k2_nat_1(np__4,B)
              & A != k1_nat_1(k2_nat_1(np__4,B),np__1)
              & A != k1_nat_1(k2_nat_1(np__4,B),np__2)
              & A != k1_nat_1(k2_nat_1(np__4,B),np__3) ) ) ) )).

fof(t4_scheme1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( A != k2_nat_1(np__5,B)
              & A != k1_nat_1(k2_nat_1(np__5,B),np__1)
              & A != k1_nat_1(k2_nat_1(np__5,B),np__2)
              & A != k1_nat_1(k2_nat_1(np__5,B),np__3)
              & A != k1_nat_1(k2_nat_1(np__5,B),np__4) ) ) ) )).

fof(s1_scheme1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & r1_xreal_0(A,B)
            & p1_s1_scheme1(k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B)) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers)
        & m1_seqm_3(A,f1_s1_scheme1)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s1_scheme1(k2_seq_1(k5_numbers,k1_numbers,A,B)) )
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ( C = k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B)
                     => p1_s1_scheme1(C) ) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,f1_s1_scheme1,B) != k2_seq_1(k5_numbers,k1_numbers,A,C) ) ) ) ) )).

fof(s2_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__2,B)) = f1_s2_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__2,B),np__1)) = f2_s2_scheme1(B) ) ) ) )).

fof(s3_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__3,B)) = f1_s3_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__3,B),np__1)) = f2_s3_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__3,B),np__2)) = f3_s3_scheme1(B) ) ) ) )).

fof(s4_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__4,B)) = f1_s4_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__1)) = f2_s4_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__2)) = f3_s4_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__4,B),np__3)) = f4_s4_scheme1(B) ) ) ) )).

fof(s5_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(np__5,B)) = f1_s5_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__1)) = f2_s5_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__2)) = f3_s5_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__3)) = f4_s5_scheme1(B)
            & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(k2_nat_1(np__5,B),np__4)) = f5_s5_scheme1(B) ) ) ) )).

fof(s6_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s6_scheme1)
       => ~ ( p1_s6_scheme1(A)
            & p2_s6_scheme1(A) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s6_scheme1,f2_s6_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s6_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s6_scheme1,f2_s6_scheme1,A))
            <=> ( p1_s6_scheme1(B)
                | p2_s6_scheme1(B) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s6_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s6_scheme1,f2_s6_scheme1,A))
             => ( ( p1_s6_scheme1(B)
                 => k1_funct_1(A,B) = f3_s6_scheme1(B) )
                & ( p2_s6_scheme1(B)
                 => k1_funct_1(A,B) = f4_s6_scheme1(B) ) ) ) ) ) )).

fof(s7_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s7_scheme1)
       => ( ( p1_s7_scheme1(A)
            & p2_s7_scheme1(A) )
         => f3_s7_scheme1(A) = f4_s7_scheme1(A) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s7_scheme1,f2_s7_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s7_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s7_scheme1,f2_s7_scheme1,A))
            <=> ( p1_s7_scheme1(B)
                | p2_s7_scheme1(B) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s7_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s7_scheme1,f2_s7_scheme1,A))
             => ( ( p1_s7_scheme1(B)
                 => k1_funct_1(A,B) = f3_s7_scheme1(B) )
                & ( p2_s7_scheme1(B)
                 => k1_funct_1(A,B) = f4_s7_scheme1(B) ) ) ) ) ) )).

fof(s8_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & m2_relset_1(A,f1_s8_scheme1,f2_s8_scheme1)
      & v1_partfun1(A,f1_s8_scheme1,f2_s8_scheme1)
      & ! [B] :
          ( m1_subset_1(B,f1_s8_scheme1)
         => ( r2_hidden(B,k4_relset_1(f1_s8_scheme1,f2_s8_scheme1,A))
           => ( ( p1_s8_scheme1(B)
               => k1_funct_1(A,B) = f3_s8_scheme1(B) )
              & ( ~ p1_s8_scheme1(B)
               => k1_funct_1(A,B) = f4_s8_scheme1(B) ) ) ) ) ) )).

fof(s9_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s9_scheme1)
       => ( ~ ( p1_s9_scheme1(A)
              & p2_s9_scheme1(A) )
          & ~ ( p1_s9_scheme1(A)
              & p3_s9_scheme1(A) )
          & ~ ( p2_s9_scheme1(A)
              & p3_s9_scheme1(A) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s9_scheme1,f2_s9_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s9_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s9_scheme1,f2_s9_scheme1,A))
            <=> ~ ( ~ p1_s9_scheme1(B)
                  & ~ p2_s9_scheme1(B)
                  & ~ p3_s9_scheme1(B) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s9_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s9_scheme1,f2_s9_scheme1,A))
             => ( ( p1_s9_scheme1(B)
                 => k1_funct_1(A,B) = f3_s9_scheme1(B) )
                & ( p2_s9_scheme1(B)
                 => k1_funct_1(A,B) = f4_s9_scheme1(B) )
                & ( p3_s9_scheme1(B)
                 => k1_funct_1(A,B) = f5_s9_scheme1(B) ) ) ) ) ) )).

fof(s10_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s10_scheme1)
       => ( ( ( p1_s10_scheme1(A)
              & p2_s10_scheme1(A) )
           => f3_s10_scheme1(A) = f4_s10_scheme1(A) )
          & ( ( p1_s10_scheme1(A)
              & p3_s10_scheme1(A) )
           => f3_s10_scheme1(A) = f5_s10_scheme1(A) )
          & ( ( p2_s10_scheme1(A)
              & p3_s10_scheme1(A) )
           => f4_s10_scheme1(A) = f5_s10_scheme1(A) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s10_scheme1,f2_s10_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s10_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s10_scheme1,f2_s10_scheme1,A))
            <=> ~ ( ~ p1_s10_scheme1(B)
                  & ~ p2_s10_scheme1(B)
                  & ~ p3_s10_scheme1(B) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s10_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s10_scheme1,f2_s10_scheme1,A))
             => ( ( p1_s10_scheme1(B)
                 => k1_funct_1(A,B) = f3_s10_scheme1(B) )
                & ( p2_s10_scheme1(B)
                 => k1_funct_1(A,B) = f4_s10_scheme1(B) )
                & ( p3_s10_scheme1(B)
                 => k1_funct_1(A,B) = f5_s10_scheme1(B) ) ) ) ) ) )).

fof(s11_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s11_scheme1)
       => ( ~ ( p1_s11_scheme1(A)
              & p2_s11_scheme1(A) )
          & ~ ( p1_s11_scheme1(A)
              & p3_s11_scheme1(A) )
          & ~ ( p1_s11_scheme1(A)
              & p4_s11_scheme1(A) )
          & ~ ( p2_s11_scheme1(A)
              & p3_s11_scheme1(A) )
          & ~ ( p2_s11_scheme1(A)
              & p4_s11_scheme1(A) )
          & ~ ( p3_s11_scheme1(A)
              & p4_s11_scheme1(A) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s11_scheme1,f2_s11_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s11_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s11_scheme1,f2_s11_scheme1,A))
            <=> ~ ( ~ p1_s11_scheme1(B)
                  & ~ p2_s11_scheme1(B)
                  & ~ p3_s11_scheme1(B)
                  & ~ p4_s11_scheme1(B) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s11_scheme1)
           => ( r2_hidden(B,k4_relset_1(f1_s11_scheme1,f2_s11_scheme1,A))
             => ( ( p1_s11_scheme1(B)
                 => k1_funct_1(A,B) = f3_s11_scheme1(B) )
                & ( p2_s11_scheme1(B)
                 => k1_funct_1(A,B) = f4_s11_scheme1(B) )
                & ( p3_s11_scheme1(B)
                 => k1_funct_1(A,B) = f5_s11_scheme1(B) )
                & ( p4_s11_scheme1(B)
                 => k1_funct_1(A,B) = f6_s11_scheme1(B) ) ) ) ) ) )).

fof(s12_scheme1,axiom,
    ( ( ! [A] :
          ~ ( r2_hidden(A,f1_s12_scheme1)
            & p1_s12_scheme1(A)
            & p2_s12_scheme1(A) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s12_scheme1)
            & p1_s12_scheme1(A) )
         => r2_hidden(f3_s12_scheme1(A),f2_s12_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s12_scheme1)
            & p2_s12_scheme1(A) )
         => r2_hidden(f4_s12_scheme1(A),f2_s12_scheme1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s12_scheme1,f2_s12_scheme1)
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s12_scheme1,f2_s12_scheme1,A))
          <=> ( r2_hidden(B,f1_s12_scheme1)
              & ( p1_s12_scheme1(B)
                | p2_s12_scheme1(B) ) ) )
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s12_scheme1,f2_s12_scheme1,A))
           => ( ( p1_s12_scheme1(B)
               => k1_funct_1(A,B) = f3_s12_scheme1(B) )
              & ( p2_s12_scheme1(B)
               => k1_funct_1(A,B) = f4_s12_scheme1(B) ) ) ) ) )).

fof(s13_scheme1,axiom,
    ( ( ! [A] :
          ( r2_hidden(A,f1_s13_scheme1)
         => ( ~ ( p1_s13_scheme1(A)
                & p2_s13_scheme1(A) )
            & ~ ( p1_s13_scheme1(A)
                & p3_s13_scheme1(A) )
            & ~ ( p2_s13_scheme1(A)
                & p3_s13_scheme1(A) ) ) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s13_scheme1)
            & p1_s13_scheme1(A) )
         => r2_hidden(f3_s13_scheme1(A),f2_s13_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s13_scheme1)
            & p2_s13_scheme1(A) )
         => r2_hidden(f4_s13_scheme1(A),f2_s13_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s13_scheme1)
            & p3_s13_scheme1(A) )
         => r2_hidden(f5_s13_scheme1(A),f2_s13_scheme1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s13_scheme1,f2_s13_scheme1)
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s13_scheme1,f2_s13_scheme1,A))
          <=> ( r2_hidden(B,f1_s13_scheme1)
              & ~ ( ~ p1_s13_scheme1(B)
                  & ~ p2_s13_scheme1(B)
                  & ~ p3_s13_scheme1(B) ) ) )
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s13_scheme1,f2_s13_scheme1,A))
           => ( ( p1_s13_scheme1(B)
               => k1_funct_1(A,B) = f3_s13_scheme1(B) )
              & ( p2_s13_scheme1(B)
               => k1_funct_1(A,B) = f4_s13_scheme1(B) )
              & ( p3_s13_scheme1(B)
               => k1_funct_1(A,B) = f5_s13_scheme1(B) ) ) ) ) )).

fof(s14_scheme1,axiom,
    ( ( ! [A] :
          ( r2_hidden(A,f1_s14_scheme1)
         => ( ~ ( p1_s14_scheme1(A)
                & p2_s14_scheme1(A) )
            & ~ ( p1_s14_scheme1(A)
                & p3_s14_scheme1(A) )
            & ~ ( p1_s14_scheme1(A)
                & p4_s14_scheme1(A) )
            & ~ ( p2_s14_scheme1(A)
                & p3_s14_scheme1(A) )
            & ~ ( p2_s14_scheme1(A)
                & p4_s14_scheme1(A) )
            & ~ ( p3_s14_scheme1(A)
                & p4_s14_scheme1(A) ) ) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s14_scheme1)
            & p1_s14_scheme1(A) )
         => r2_hidden(f3_s14_scheme1(A),f2_s14_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s14_scheme1)
            & p2_s14_scheme1(A) )
         => r2_hidden(f4_s14_scheme1(A),f2_s14_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s14_scheme1)
            & p3_s14_scheme1(A) )
         => r2_hidden(f5_s14_scheme1(A),f2_s14_scheme1) )
      & ! [A] :
          ( ( r2_hidden(A,f1_s14_scheme1)
            & p4_s14_scheme1(A) )
         => r2_hidden(f6_s14_scheme1(A),f2_s14_scheme1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,f1_s14_scheme1,f2_s14_scheme1)
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s14_scheme1,f2_s14_scheme1,A))
          <=> ( r2_hidden(B,f1_s14_scheme1)
              & ~ ( ~ p1_s14_scheme1(B)
                  & ~ p2_s14_scheme1(B)
                  & ~ p3_s14_scheme1(B)
                  & ~ p4_s14_scheme1(B) ) ) )
        & ! [B] :
            ( r2_hidden(B,k4_relset_1(f1_s14_scheme1,f2_s14_scheme1,A))
           => ( ( p1_s14_scheme1(B)
               => k1_funct_1(A,B) = f3_s14_scheme1(B) )
              & ( p2_s14_scheme1(B)
               => k1_funct_1(A,B) = f4_s14_scheme1(B) )
              & ( p3_s14_scheme1(B)
               => k1_funct_1(A,B) = f5_s14_scheme1(B) )
              & ( p4_s14_scheme1(B)
               => k1_funct_1(A,B) = f6_s14_scheme1(B) ) ) ) ) )).

fof(s15_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s15_scheme1)
       => ! [B] :
            ( m1_subset_1(B,f2_s15_scheme1)
           => ~ ( p1_s15_scheme1(A,B)
                & p2_s15_scheme1(A,B) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s15_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s15_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1,A))
                <=> ( p1_s15_scheme1(B,C)
                    | p2_s15_scheme1(B,C) ) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s15_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s15_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s15_scheme1,f2_s15_scheme1),f3_s15_scheme1,A))
                 => ( ( p1_s15_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f4_s15_scheme1(B,C) )
                    & ( p2_s15_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f5_s15_scheme1(B,C) ) ) ) ) ) ) )).

fof(s16_scheme1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s16_scheme1)
       => ! [B] :
            ( m1_subset_1(B,f2_s16_scheme1)
           => ( ~ ( p1_s16_scheme1(A,B)
                  & p2_s16_scheme1(A,B) )
              & ~ ( p1_s16_scheme1(A,B)
                  & p3_s16_scheme1(A,B) )
              & ~ ( p2_s16_scheme1(A,B)
                  & p3_s16_scheme1(A,B) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s16_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s16_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1,A))
                <=> ~ ( ~ p1_s16_scheme1(B,C)
                      & ~ p2_s16_scheme1(B,C)
                      & ~ p3_s16_scheme1(B,C) ) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s16_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s16_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s16_scheme1,f2_s16_scheme1),f3_s16_scheme1,A))
                 => ( ( p1_s16_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f4_s16_scheme1(B,C) )
                    & ( p2_s16_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f5_s16_scheme1(B,C) )
                    & ( p3_s16_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f6_s16_scheme1(B,C) ) ) ) ) ) ) )).

fof(s17_scheme1,axiom,
    ( ( ! [A,B] :
          ~ ( r2_hidden(A,f1_s17_scheme1)
            & r2_hidden(B,f2_s17_scheme1)
            & p1_s17_scheme1(A,B)
            & p2_s17_scheme1(A,B) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s17_scheme1)
            & r2_hidden(B,f2_s17_scheme1)
            & p1_s17_scheme1(A,B) )
         => r2_hidden(f4_s17_scheme1(A,B),f3_s17_scheme1) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s17_scheme1)
            & r2_hidden(B,f2_s17_scheme1)
            & p2_s17_scheme1(A,B) )
         => r2_hidden(f5_s17_scheme1(A,B),f3_s17_scheme1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1)
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1,A))
          <=> ( r2_hidden(B,f1_s17_scheme1)
              & r2_hidden(C,f2_s17_scheme1)
              & ( p1_s17_scheme1(B,C)
                | p2_s17_scheme1(B,C) ) ) )
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s17_scheme1,f2_s17_scheme1),f3_s17_scheme1,A))
           => ( ( p1_s17_scheme1(B,C)
               => k1_funct_1(A,k4_tarski(B,C)) = f4_s17_scheme1(B,C) )
              & ( p2_s17_scheme1(B,C)
               => k1_funct_1(A,k4_tarski(B,C)) = f5_s17_scheme1(B,C) ) ) ) ) )).

fof(s18_scheme1,axiom,
    ( ( ! [A,B] :
          ( ( r2_hidden(A,f1_s18_scheme1)
            & r2_hidden(B,f2_s18_scheme1) )
         => ( ~ ( p1_s18_scheme1(A,B)
                & p2_s18_scheme1(A,B) )
            & ~ ( p1_s18_scheme1(A,B)
                & p3_s18_scheme1(A,B) )
            & ~ ( p2_s18_scheme1(A,B)
                & p3_s18_scheme1(A,B) ) ) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s18_scheme1)
            & r2_hidden(B,f2_s18_scheme1)
            & p1_s18_scheme1(A,B) )
         => r2_hidden(f4_s18_scheme1(A,B),f3_s18_scheme1) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s18_scheme1)
            & r2_hidden(B,f2_s18_scheme1)
            & p2_s18_scheme1(A,B) )
         => r2_hidden(f5_s18_scheme1(A,B),f3_s18_scheme1) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s18_scheme1)
            & r2_hidden(B,f2_s18_scheme1)
            & p3_s18_scheme1(A,B) )
         => r2_hidden(f6_s18_scheme1(A,B),f3_s18_scheme1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & m2_relset_1(A,k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1)
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1,A))
          <=> ( r2_hidden(B,f1_s18_scheme1)
              & r2_hidden(C,f2_s18_scheme1)
              & ~ ( ~ p1_s18_scheme1(B,C)
                  & ~ p2_s18_scheme1(B,C)
                  & ~ p3_s18_scheme1(B,C) ) ) )
        & ! [B,C] :
            ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s18_scheme1,f2_s18_scheme1),f3_s18_scheme1,A))
           => ( ( p1_s18_scheme1(B,C)
               => k1_funct_1(A,k4_tarski(B,C)) = f4_s18_scheme1(B,C) )
              & ( p2_s18_scheme1(B,C)
               => k1_funct_1(A,k4_tarski(B,C)) = f5_s18_scheme1(B,C) )
              & ( p3_s18_scheme1(B,C)
               => k1_funct_1(A,k4_tarski(B,C)) = f6_s18_scheme1(B,C) ) ) ) ) )).

fof(s19_scheme1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s19_scheme1)
         => ( ~ ( p1_s19_scheme1(A)
                & p2_s19_scheme1(A) )
            & ~ ( p1_s19_scheme1(A)
                & p3_s19_scheme1(A) )
            & ~ ( p2_s19_scheme1(A)
                & p3_s19_scheme1(A) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s19_scheme1)
         => ~ ( ~ p1_s19_scheme1(A)
              & ~ p2_s19_scheme1(A)
              & ~ p3_s19_scheme1(A) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s19_scheme1,f2_s19_scheme1)
        & m2_relset_1(A,f1_s19_scheme1,f2_s19_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s19_scheme1)
           => ( ( p1_s19_scheme1(B)
               => k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f3_s19_scheme1(B) )
              & ( p2_s19_scheme1(B)
               => k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f4_s19_scheme1(B) )
              & ( p3_s19_scheme1(B)
               => k8_funct_2(f1_s19_scheme1,f2_s19_scheme1,A,B) = f5_s19_scheme1(B) ) ) ) ) )).

fof(s20_scheme1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s20_scheme1)
         => ( ~ ( p1_s20_scheme1(A)
                & p2_s20_scheme1(A) )
            & ~ ( p1_s20_scheme1(A)
                & p3_s20_scheme1(A) )
            & ~ ( p1_s20_scheme1(A)
                & p4_s20_scheme1(A) )
            & ~ ( p2_s20_scheme1(A)
                & p3_s20_scheme1(A) )
            & ~ ( p2_s20_scheme1(A)
                & p4_s20_scheme1(A) )
            & ~ ( p3_s20_scheme1(A)
                & p4_s20_scheme1(A) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s20_scheme1)
         => ~ ( ~ p1_s20_scheme1(A)
              & ~ p2_s20_scheme1(A)
              & ~ p3_s20_scheme1(A)
              & ~ p4_s20_scheme1(A) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s20_scheme1,f2_s20_scheme1)
        & m2_relset_1(A,f1_s20_scheme1,f2_s20_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s20_scheme1)
           => ( ( p1_s20_scheme1(B)
               => k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f3_s20_scheme1(B) )
              & ( p2_s20_scheme1(B)
               => k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f4_s20_scheme1(B) )
              & ( p3_s20_scheme1(B)
               => k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f5_s20_scheme1(B) )
              & ( p4_s20_scheme1(B)
               => k8_funct_2(f1_s20_scheme1,f2_s20_scheme1,A,B) = f6_s20_scheme1(B) ) ) ) ) )).

fof(s21_scheme1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1)
      & m2_relset_1(A,k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1)
      & ! [B] :
          ( m1_subset_1(B,f1_s21_scheme1)
         => ! [C] :
              ( m1_subset_1(C,f2_s21_scheme1)
             => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s21_scheme1,f2_s21_scheme1),f3_s21_scheme1,A))
               => ( ( p1_s21_scheme1(B,C)
                   => k1_funct_1(A,k4_tarski(B,C)) = f4_s21_scheme1(B,C) )
                  & ( ~ p1_s21_scheme1(B,C)
                   => k1_funct_1(A,k4_tarski(B,C)) = f5_s21_scheme1(B,C) ) ) ) ) ) ) )).

fof(s22_scheme1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s22_scheme1)
         => ! [B] :
              ( m1_subset_1(B,f2_s22_scheme1)
             => ( ~ ( p1_s22_scheme1(A,B)
                    & p2_s22_scheme1(A,B) )
                & ~ ( p1_s22_scheme1(A,B)
                    & p3_s22_scheme1(A,B) )
                & ~ ( p2_s22_scheme1(A,B)
                    & p3_s22_scheme1(A,B) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s22_scheme1)
         => ! [B] :
              ( m1_subset_1(B,f2_s22_scheme1)
             => ~ ( ~ p1_s22_scheme1(A,B)
                  & ~ p2_s22_scheme1(A,B)
                  & ~ p3_s22_scheme1(A,B) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1)
        & m2_relset_1(A,k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1)
        & ! [B] :
            ( m1_subset_1(B,f1_s22_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s22_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1,A))
                <=> ~ ( ~ p1_s22_scheme1(B,C)
                      & ~ p2_s22_scheme1(B,C)
                      & ~ p3_s22_scheme1(B,C) ) ) ) )
        & ! [B] :
            ( m1_subset_1(B,f1_s22_scheme1)
           => ! [C] :
                ( m1_subset_1(C,f2_s22_scheme1)
               => ( r2_hidden(k4_tarski(B,C),k4_relset_1(k2_zfmisc_1(f1_s22_scheme1,f2_s22_scheme1),f3_s22_scheme1,A))
                 => ( ( p1_s22_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f4_s22_scheme1(B,C) )
                    & ( p2_s22_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f5_s22_scheme1(B,C) )
                    & ( p3_s22_scheme1(B,C)
                     => k1_funct_1(A,k4_tarski(B,C)) = f6_s22_scheme1(B,C) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------