SET007 Axioms: SET007+656.ax


%------------------------------------------------------------------------------
% File     : SET007+656 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Segre's Product of Partial Line Spaces
% 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    : pencil_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   80 (   0 unit)
%            Number of atoms       :  552 (  41 equality)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  591 ( 119 ~  ;   2  |; 254  &)
%                                         (  30 <=>; 186 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   0 propositional; 1-3 arity)
%            Number of functors    :   30 (   5 constant; 0-4 arity)
%            Number of variables   :  216 (   0 singleton; 192 !;  24 ?)
%            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_pencil_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v1_pre_topc(A)
      & ~ v3_pencil_1(A)
      & ~ v4_pencil_1(A)
      & v5_pencil_1(A)
      & v6_pencil_1(A)
      & ~ v7_pencil_1(A)
      & v8_pencil_1(A) ) )).

fof(rc2_pencil_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v1_pre_topc(A)
      & ~ v3_pencil_1(A)
      & ~ v4_pencil_1(A)
      & v5_pencil_1(A)
      & v6_pencil_1(A)
      & v7_pencil_1(A)
      & v8_pencil_1(A) ) )).

fof(fc1_pencil_1,axiom,(
    ! [A] :
      ( ( ~ v3_pencil_1(A)
        & l1_pre_topc(A) )
     => ~ v1_xboole_0(u1_pre_topc(A)) ) )).

fof(cc1_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v11_pencil_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_pralg_1(A) ) ) )).

fof(rc3_pencil_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v2_pralg_1(B)
      & v11_pencil_1(B) ) )).

fof(rc4_pencil_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_pralg_1(A)
      & v11_pencil_1(A) ) )).

fof(cc2_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v14_pencil_1(A) )
     => ( v1_relat_1(A)
        & v4_waybel_3(A) ) ) )).

fof(cc3_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v15_pencil_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v4_waybel_3(A)
        & v2_pralg_1(A)
        & v11_pencil_1(A) ) ) )).

fof(cc4_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v11_pencil_1(A)
        & v15_pencil_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_pralg_1(A)
        & v11_pencil_1(A)
        & v12_pencil_1(A) ) ) )).

fof(cc5_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v11_pencil_1(A)
        & v15_pencil_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v4_waybel_3(A)
        & v2_pralg_1(A)
        & v11_pencil_1(A)
        & v14_pencil_1(A) ) ) )).

fof(rc5_pencil_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v4_waybel_3(B)
      & v2_pralg_1(B)
      & v11_pencil_1(B)
      & v12_pencil_1(B)
      & v14_pencil_1(B)
      & v15_pencil_1(B) ) )).

fof(fc2_pencil_1,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v1_relat_1(k1_pzfmisc1(A,B))
        & v2_relat_1(k1_pzfmisc1(A,B))
        & v1_funct_1(k1_pzfmisc1(A,B))
        & v1_pre_circ(k1_pzfmisc1(A,B),A)
        & v13_pencil_1(k1_pzfmisc1(A,B)) ) ) )).

fof(fc3_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A) )
     => ( v1_relat_1(k1_pzfmisc1(A,B))
        & v2_relat_1(k1_pzfmisc1(A,B))
        & v1_funct_1(k1_pzfmisc1(A,B))
        & v1_pre_circ(k1_pzfmisc1(A,B),A)
        & v13_pencil_1(k1_pzfmisc1(A,B))
        & v16_pencil_1(k1_pzfmisc1(A,B),A) ) ) )).

fof(rc6_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_waybel_3(B)
        & v2_pralg_1(B)
        & m1_pboole(B,A) )
     => ? [C] :
          ( m4_pboole(C,A,k12_pralg_1(A,B))
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C)
          & v13_pencil_1(C)
          & v16_pencil_1(C,A) ) ) )).

fof(rc7_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v2_pralg_1(B)
        & v14_pencil_1(B)
        & m1_pboole(B,A) )
     => ? [C] :
          ( m4_pboole(C,A,k12_pralg_1(A,B))
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C)
          & ~ v13_pencil_1(C)
          & v16_pencil_1(C,A) ) ) )).

fof(rc8_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_pboole(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & ~ v13_pencil_1(B)
          & v16_pencil_1(B,A) ) ) )).

fof(cc6_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ( ( ~ v13_pencil_1(B)
              & v16_pencil_1(B,A) )
           => v2_relat_1(B) ) ) ) )).

fof(fc4_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v13_pencil_1(B)
        & v16_pencil_1(B,A)
        & m1_pboole(B,A) )
     => ( ~ v1_xboole_0(k4_card_3(B))
        & v1_fraenkel(k4_card_3(B))
        & ~ v1_realset1(k4_card_3(B)) ) ) )).

fof(t1_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k4_card_3(A) = k4_card_3(B)
              & v2_relat_1(A) )
           => v2_relat_1(B) ) ) ) )).

fof(t2_pencil_1,axiom,(
    ! [A] :
      ( r1_tarski(np__2,k1_card_1(A))
    <=> ? [B,C] :
          ( r2_hidden(B,A)
          & r2_hidden(C,A)
          & B != C ) ) )).

fof(t3_pencil_1,axiom,(
    ! [A] :
      ( r1_tarski(np__2,k1_card_1(A))
     => ! [B] :
        ? [C] :
          ( r2_hidden(C,A)
          & B != C ) ) )).

fof(t4_pencil_1,axiom,(
    ! [A] :
      ( r1_tarski(np__2,k1_card_1(A))
    <=> ~ v1_realset1(A) ) )).

fof(t5_pencil_1,axiom,(
    ! [A] :
      ( r1_tarski(np__3,k1_card_1(A))
    <=> ? [B,C,D] :
          ( r2_hidden(B,A)
          & r2_hidden(C,A)
          & r2_hidden(D,A)
          & B != C
          & B != D
          & C != D ) ) )).

fof(t6_pencil_1,axiom,(
    ! [A] :
      ( r1_tarski(np__3,k1_card_1(A))
     => ! [B,C] :
        ? [D] :
          ( r2_hidden(D,A)
          & B != D
          & C != D ) ) )).

fof(d1_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_pencil_1(A,B,C)
              <=> ~ ( B != C
                    & ! [D] :
                        ( m1_subset_1(D,u1_pre_topc(A))
                       => ~ r1_tarski(k2_tarski(B,C),D) ) ) ) ) ) ) )).

fof(d2_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_pencil_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_pre_topc(A))
               => ( r1_tarski(np__2,k1_card_1(k3_xboole_0(C,B)))
                 => r1_tarski(C,B) ) ) ) ) ) )).

fof(d3_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_pencil_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,B)
                        & r2_hidden(D,B) )
                     => r1_pencil_1(A,C,D) ) ) ) ) ) ) )).

fof(d4_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v3_pencil_1(A)
      <=> v1_xboole_0(u1_pre_topc(A)) ) ) )).

fof(d5_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v4_pencil_1(A)
      <=> m1_subset_1(u1_struct_0(A),u1_pre_topc(A)) ) ) )).

fof(d6_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v5_pencil_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_pre_topc(A))
           => r1_tarski(np__2,k1_card_1(B)) ) ) ) )).

fof(d7_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v6_pencil_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_pre_topc(A))
           => ! [C] :
                ( m1_subset_1(C,u1_pre_topc(A))
               => ( r1_tarski(np__2,k1_card_1(k3_xboole_0(B,C)))
                 => B = C ) ) ) ) ) )).

fof(d8_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v7_pencil_1(A)
      <=> ~ ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => r1_pencil_1(A,B,C) ) ) ) ) )).

fof(d9_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v8_pencil_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ? [C] :
                ( m1_subset_1(C,u1_pre_topc(A))
                & r2_hidden(B,C) ) ) ) ) )).

fof(d10_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v9_pencil_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m2_finseq_1(D,u1_struct_0(A))
                    & B = k1_funct_1(D,np__1)
                    & C = k1_funct_1(D,k3_finseq_1(D))
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r1_xreal_0(np__1,E)
                         => ( r1_xreal_0(k3_finseq_1(D),E)
                            | ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ! [G] :
                                    ( m1_subset_1(G,u1_struct_0(A))
                                   => ( ( F = k1_funct_1(D,E)
                                        & G = k1_funct_1(D,k1_nat_1(E,np__1)) )
                                     => r1_pencil_1(A,F,G) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d11_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v10_pencil_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( v1_pencil_1(C,A)
                    & v2_pencil_1(C,A)
                    & ! [D] :
                        ( m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( C = k1_funct_1(D,np__1)
                            & r2_hidden(B,k1_funct_1(D,k3_finseq_1(D)))
                            & ! [E] :
                                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                               => ( r2_hidden(E,k2_relat_1(D))
                                 => ( v1_pencil_1(E,A)
                                    & v2_pencil_1(E,A) ) ) )
                            & ! [E] :
                                ( m2_subset_1(E,k1_numbers,k5_numbers)
                               => ( r1_xreal_0(np__1,E)
                                 => ( r1_xreal_0(k3_finseq_1(D),E)
                                    | r1_tarski(np__2,k1_card_1(k3_xboole_0(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))))) ) ) ) ) ) ) ) ) ) ) )).

fof(d12_pencil_1,axiom,(
    ! [A] :
      ( ( v8_pencil_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_pencil_1(A,B,C)
              <=> ? [D] :
                    ( m1_subset_1(D,u1_pre_topc(A))
                    & r1_tarski(k2_tarski(B,C),D) ) ) ) ) ) )).

fof(d13_pencil_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v11_pencil_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => l1_pre_topc(B) ) ) ) )).

fof(d14_pencil_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v12_pencil_1(A)
      <=> ! [B] :
            ( l1_pre_topc(B)
           => ~ ( r2_hidden(B,k2_relat_1(A))
                & v3_pencil_1(B) ) ) ) ) )).

fof(d15_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v11_pencil_1(A) )
     => ( v12_pencil_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => ( ~ v3_pencil_1(B)
              & l1_pre_topc(B) ) ) ) ) )).

fof(d16_pencil_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v13_pencil_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => v1_realset1(B) ) ) ) )).

fof(d17_pencil_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v14_pencil_1(A)
      <=> ! [B] :
            ( l1_struct_0(B)
           => ~ ( r2_hidden(B,k2_relat_1(A))
                & v3_realset2(B) ) ) ) ) )).

fof(d18_pencil_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_pralg_1(A) )
     => ( v14_pencil_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => ( ~ v3_realset2(B)
              & l1_struct_0(B) ) ) ) ) )).

fof(d19_pencil_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v15_pencil_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => ( ~ v3_struct_0(B)
              & ~ v3_pencil_1(B)
              & ~ v4_pencil_1(B)
              & v5_pencil_1(B)
              & v6_pencil_1(B)
              & l1_pre_topc(B) ) ) ) ) )).

fof(d20_pencil_1,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v16_pencil_1(B,A)
      <=> ? [C] :
            ( m1_subset_1(C,A)
            & ! [D] :
                ( m1_subset_1(D,A)
               => ( C != D
                 => ( ~ v1_xboole_0(k1_funct_1(B,D))
                    & v1_realset1(k1_funct_1(B,D)) ) ) ) ) ) ) )).

fof(t9_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ~ v1_realset1(D)
                 => ~ v13_pencil_1(k2_polynom1(A,B,C,D)) ) ) ) ) )).

fof(t10_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C,D] : v16_pencil_1(k2_polynom1(A,k1_pzfmisc1(A,B),C,D),A) ) ) )).

fof(t11_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v2_pralg_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m2_pboole(C,A,k12_pralg_1(A,B))
             => m4_pboole(k1_pzfmisc1(A,C),A,k12_pralg_1(A,B)) ) ) ) )).

fof(d21_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( C = k3_pencil_1(A,B)
              <=> ~ v1_realset1(k1_funct_1(B,C)) ) ) ) ) )).

fof(t12_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( C != k3_pencil_1(A,B)
               => ( ~ v1_xboole_0(k1_funct_1(B,C))
                  & v1_realset1(k1_funct_1(B,C)) ) ) ) ) ) )).

fof(t13_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ( r1_tarski(np__2,k1_card_1(k4_card_3(B)))
          <=> ( v2_relat_1(B)
              & ~ v13_pencil_1(B) ) ) ) ) )).

fof(d22_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k4_card_3(k12_pralg_1(A,B)))))
             => ( C = k4_pencil_1(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ? [E] :
                        ( v16_pencil_1(E,A)
                        & m4_pboole(E,A,k12_pralg_1(A,B))
                        & D = k4_card_3(E)
                        & ? [F] :
                            ( m1_subset_1(F,A)
                            & m1_subset_1(k1_funct_1(E,F),u1_pre_topc(k1_pencil_1(A,B,F))) ) ) ) ) ) ) ) )).

fof(d23_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => k5_pencil_1(A,B) = g1_pre_topc(k4_card_3(k12_pralg_1(A,B)),k4_pencil_1(A,B)) ) ) )).

fof(t14_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k5_pencil_1(A,B)))
             => m1_pboole(C,A) ) ) ) )).

fof(t15_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ~ ( ~ ! [C] :
                    ( m1_subset_1(C,A)
                   => v3_pencil_1(k1_pencil_1(A,B,C)) )
              & v3_pencil_1(k5_pencil_1(A,B)) ) ) ) )).

fof(t16_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ~ ( ! [C] :
                  ( m1_subset_1(C,A)
                 => ( ~ v4_pencil_1(k1_pencil_1(A,B,C))
                    & ~ ! [D] :
                          ( m1_subset_1(D,A)
                         => v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
              & v4_pencil_1(k5_pencil_1(A,B)) ) ) ) )).

fof(t17_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v5_pencil_1(k1_pencil_1(A,B,C))
                  & ~ ! [D] :
                        ( m1_subset_1(D,A)
                       => v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
           => v5_pencil_1(k5_pencil_1(A,B)) ) ) ) )).

fof(t18_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v11_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v6_pencil_1(k1_pencil_1(A,B,C))
                  & v5_pencil_1(k1_pencil_1(A,B,C))
                  & ~ ! [D] :
                        ( m1_subset_1(D,A)
                       => v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
           => v6_pencil_1(k5_pencil_1(A,B)) ) ) ) )).

fof(t19_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_realset1(B)
           => ( v2_pencil_1(B,A)
              & v1_pencil_1(B,A) ) ) ) ) )).

fof(t20_pencil_1,axiom,(
    ! [A] :
      ( ( v6_pencil_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_pre_topc(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = B
               => v1_pencil_1(C,A) ) ) ) ) )).

fof(t21_pencil_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,u1_pre_topc(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = B
               => v2_pencil_1(C,A) ) ) ) ) )).

fof(t22_pencil_1,axiom,(
    ! [A] :
      ( ( ~ v3_pencil_1(A)
        & l1_pre_topc(A) )
     => v1_pencil_1(k2_pre_topc(A),A) ) )).

fof(t23_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_pboole(C,A)
             => ! [D] :
                  ( m1_pboole(D,A)
                 => ( ( r2_hidden(C,k4_card_3(B))
                      & r2_hidden(D,k4_card_3(B)) )
                   => ! [E] :
                        ( E != k3_pencil_1(A,B)
                       => k1_funct_1(C,E) = k1_funct_1(D,E) ) ) ) ) ) ) )).

fof(t24_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_pre_topc(k6_pencil_1(A,B)))
            <=> ? [D] :
                  ( ~ v13_pencil_1(D)
                  & v16_pencil_1(D,A)
                  & m4_pboole(D,A,k12_pralg_1(A,B))
                  & C = k4_card_3(D)
                  & m1_subset_1(k1_funct_1(D,k3_pencil_1(A,D)),u1_pre_topc(k2_pencil_1(A,B,k3_pencil_1(A,D)))) ) ) ) ) )).

fof(t25_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_pboole(C,A)
             => ( m1_subset_1(C,u1_struct_0(k6_pencil_1(A,B)))
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k2_pencil_1(A,B,D)))
                       => m1_subset_1(k2_polynom1(A,C,D,E),u1_struct_0(k6_pencil_1(A,B))) ) ) ) ) ) ) )).

fof(t26_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( ~ v13_pencil_1(C)
                & v16_pencil_1(C,A)
                & m1_pboole(C,A) )
             => ( r1_tarski(np__2,k1_card_1(k3_xboole_0(k4_card_3(B),k4_card_3(C))))
               => ( k3_pencil_1(A,B) = k3_pencil_1(A,C)
                  & ! [D] :
                      ( D != k3_pencil_1(A,B)
                     => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ) )).

fof(t27_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v13_pencil_1(B)
            & v16_pencil_1(B,A)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ~ v1_realset1(C)
             => ( v16_pencil_1(k2_polynom1(A,B,k3_pencil_1(A,B),C),A)
                & ~ v13_pencil_1(k2_polynom1(A,B,k3_pencil_1(A,B),C)) ) ) ) ) )).

fof(t28_pencil_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_pencil_1(A)
        & v6_pencil_1(A)
        & v8_pencil_1(A)
        & l1_pre_topc(A) )
     => ( v10_pencil_1(A)
       => v9_pencil_1(A) ) ) )).

fof(t29_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
             => ( ( ~ v1_realset1(C)
                  & v2_pencil_1(C,k6_pencil_1(A,B))
                  & v1_pencil_1(C,k6_pencil_1(A,B)) )
              <=> ? [D] :
                    ( ~ v13_pencil_1(D)
                    & v16_pencil_1(D,A)
                    & m4_pboole(D,A,k12_pralg_1(A,B))
                    & C = k4_card_3(D)
                    & ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D)))))
                       => ( E = k1_funct_1(D,k3_pencil_1(A,D))
                         => ( v2_pencil_1(E,k2_pencil_1(A,B,k3_pencil_1(A,D)))
                            & v1_pencil_1(E,k2_pencil_1(A,B,k3_pencil_1(A,D))) ) ) ) ) ) ) ) ) )).

fof(dt_k1_pencil_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v11_pencil_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => l1_pre_topc(k1_pencil_1(A,B,C)) ) )).

fof(redefinition_k1_pencil_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v11_pencil_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => k1_pencil_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k2_pencil_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => ( ~ v3_struct_0(k2_pencil_1(A,B,C))
        & ~ v3_pencil_1(k2_pencil_1(A,B,C))
        & ~ v4_pencil_1(k2_pencil_1(A,B,C))
        & v5_pencil_1(k2_pencil_1(A,B,C))
        & v6_pencil_1(k2_pencil_1(A,B,C))
        & l1_pre_topc(k2_pencil_1(A,B,C)) ) ) )).

fof(redefinition_k2_pencil_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => k2_pencil_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k3_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v13_pencil_1(B)
        & v16_pencil_1(B,A)
        & m1_pboole(B,A) )
     => m1_subset_1(k3_pencil_1(A,B),A) ) )).

fof(dt_k4_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_waybel_3(B)
        & v11_pencil_1(B)
        & m1_pboole(B,A) )
     => m1_subset_1(k4_pencil_1(A,B),k1_zfmisc_1(k1_zfmisc_1(k4_card_3(k12_pralg_1(A,B))))) ) )).

fof(dt_k5_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_waybel_3(B)
        & v11_pencil_1(B)
        & m1_pboole(B,A) )
     => ( ~ v3_struct_0(k5_pencil_1(A,B))
        & l1_pre_topc(k5_pencil_1(A,B)) ) ) )).

fof(dt_k6_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A) )
     => ( ~ v3_struct_0(k6_pencil_1(A,B))
        & ~ v3_pencil_1(k6_pencil_1(A,B))
        & ~ v4_pencil_1(k6_pencil_1(A,B))
        & v5_pencil_1(k6_pencil_1(A,B))
        & v6_pencil_1(k6_pencil_1(A,B))
        & l1_pre_topc(k6_pencil_1(A,B)) ) ) )).

fof(redefinition_k6_pencil_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A) )
     => k6_pencil_1(A,B) = k5_pencil_1(A,B) ) )).

fof(t7_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( r1_tarski(np__3,k1_card_1(A))
       => ! [B] :
            ( l1_pre_topc(B)
           => ( ( u1_struct_0(B) = A
                & u1_pre_topc(B) = a_1_0_pencil_1(A) )
             => ( ~ v3_struct_0(B)
                & ~ v3_pencil_1(B)
                & ~ v4_pencil_1(B)
                & ~ v7_pencil_1(B)
                & v5_pencil_1(B)
                & v6_pencil_1(B)
                & v8_pencil_1(B) ) ) ) ) ) )).

fof(t8_pencil_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( r1_tarski(np__3,k1_card_1(A))
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(A))
           => ( k1_card_1(B) = np__2
             => ! [C] :
                  ( l1_pre_topc(C)
                 => ( ( u1_struct_0(C) = A
                      & u1_pre_topc(C) = k4_xboole_0(a_1_0_pencil_1(A),k1_tarski(B)) )
                   => ( ~ v3_struct_0(C)
                      & ~ v3_pencil_1(C)
                      & ~ v4_pencil_1(C)
                      & v7_pencil_1(C)
                      & v5_pencil_1(C)
                      & v6_pencil_1(C)
                      & v8_pencil_1(C) ) ) ) ) ) ) ) )).

fof(fraenkel_a_1_0_pencil_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_0_pencil_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(B))
            & A = C
            & np__2 = k1_card_1(C) ) ) ) )).
%------------------------------------------------------------------------------