SET007 Axioms: SET007+877.ax


%------------------------------------------------------------------------------
% File     : SET007+877 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Characterization of Collineations of the Segre Product
% Version  : [Urb08] axioms.
% English  : On the Characterization of Collineations of the Segre Product of
%            Strongly Connected Partial Linear Spaces

% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (   0 unit)
%            Number of atoms       :  555 (  74 equality)
%            Maximal formula depth :   35 (  16 average)
%            Number of connectives :  607 (  90 ~  ;   3  |; 263  &)
%                                         (  10 <=>; 241 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   0 propositional; 1-4 arity)
%            Number of functors    :   31 (   3 constant; 0-4 arity)
%            Number of variables   :  215 (   0 singleton; 206 !;   9 ?)
%            Maximal term depth    :    6 (   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_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_pencil_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v1_pencil_2(B,A,A)
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_pencil_1(A,C,D)
                   => r1_pencil_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) ) ) ) ) )).

fof(t2_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v2_pralg_1(C)
                & v14_pencil_1(C)
                & m1_pboole(C,A) )
             => ~ v3_realset2(k10_pralg_1(A,C,B)) ) ) ) )).

fof(t3_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v3_pencil_1(A)
        & v6_pencil_1(A)
        & l1_pre_topc(A) )
     => ( v10_pencil_1(A)
       => v8_pencil_1(A) ) ) )).

fof(t4_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_pencil_1(A)
        & v6_pencil_1(A)
        & l1_pre_topc(A) )
     => ( v10_pencil_1(A)
       => v9_pencil_1(A) ) ) )).

fof(t5_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v3_pencil_1(A)
        & ~ v4_pencil_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_pre_topc(A))
         => ~ ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => r2_hidden(C,B) ) ) ) )).

fof(t6_pencil_3,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)))
             => m2_pboole(C,A,k12_pralg_1(A,B)) ) ) ) )).

fof(t7_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_pralg_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => k1_funct_1(k12_pralg_1(A,B),C) = k2_pre_topc(k10_pralg_1(A,B,C)) ) ) ) )).

fof(t8_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v11_pencil_1(C)
                & v14_pencil_1(C)
                & m1_pboole(C,A) )
             => ? [D] :
                  ( ~ v13_pencil_1(D)
                  & v16_pencil_1(D,A)
                  & m4_pboole(D,A,k12_pralg_1(A,C))
                  & k3_pencil_1(A,D) = B
                  & m1_pencil_2(k4_card_3(D),A,C) ) ) ) ) )).

fof(t9_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v11_pencil_1(C)
                & v14_pencil_1(C)
                & m1_pboole(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k5_pencil_1(A,C)))
                 => ? [E] :
                      ( ~ v13_pencil_1(E)
                      & v16_pencil_1(E,A)
                      & m4_pboole(E,A,k12_pralg_1(A,C))
                      & k3_pencil_1(A,E) = B
                      & m1_pencil_2(k4_card_3(E),A,C)
                      & r2_hidden(D,k4_card_3(E)) ) ) ) ) ) )).

fof(t10_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v11_pencil_1(B)
            & v14_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( ~ v13_pencil_1(C)
                & v16_pencil_1(C,A)
                & m4_pboole(C,A,k12_pralg_1(A,B)) )
             => ( m1_pencil_2(k4_card_3(C),A,B)
               => k1_funct_1(C,k3_pencil_1(A,C)) = k2_pre_topc(k1_pencil_1(A,B,k3_pencil_1(A,C))) ) ) ) ) )).

fof(t11_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v11_pencil_1(B)
            & v14_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( ~ v13_pencil_1(C)
                & v16_pencil_1(C,A)
                & m4_pboole(C,A,k12_pralg_1(A,B)) )
             => ! [D] :
                  ( ( ~ v13_pencil_1(D)
                    & v16_pencil_1(D,A)
                    & m4_pboole(D,A,k12_pralg_1(A,B)) )
                 => ( ( m1_pencil_2(k4_card_3(C),A,B)
                      & m1_pencil_2(k4_card_3(D),A,B)
                      & k3_pencil_1(A,C) = k3_pencil_1(A,D) )
                   => ( r2_subset_1(k4_card_3(C),k4_card_3(D))
                      | r6_pboole(A,C,D) ) ) ) ) ) ) )).

fof(t12_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( ~ v13_pencil_1(C)
                & v16_pencil_1(C,A)
                & m4_pboole(C,A,k12_pralg_1(A,B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_pre_topc(k2_pencil_1(A,B,k3_pencil_1(A,C))))
                 => m1_subset_1(k4_card_3(k2_polynom1(A,C,k3_pencil_1(A,C),D)),u1_pre_topc(k6_pencil_1(A,B))) ) ) ) ) )).

fof(t13_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C)))
                 => ! [E] :
                      ( ( ~ v13_pencil_1(E)
                        & v16_pencil_1(E,A)
                        & m4_pboole(E,A,k12_pralg_1(A,B)) )
                     => ( C != k3_pencil_1(A,E)
                       => ( ~ v13_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)))
                          & v16_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A)
                          & m4_pboole(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A,k12_pralg_1(A,B)) ) ) ) ) ) ) ) )).

fof(t14_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k2_pencil_1(A,B,C))))
                 => ! [E] :
                      ( ( ~ v13_pencil_1(E)
                        & v16_pencil_1(E,A)
                        & m4_pboole(E,A,k12_pralg_1(A,B)) )
                     => m1_subset_1(k4_card_3(k2_polynom1(A,E,C,D)),k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) ) ) ) ) ) )).

fof(t15_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ~ v1_xboole_0(k1_funct_1(k1_pzfmisc1(A,B),C))
                & v1_realset1(k1_funct_1(k1_pzfmisc1(A,B),C)) ) ) ) ) )).

fof(t16_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v15_pencil_1(C)
                & m1_pboole(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_pre_topc(k2_pencil_1(A,C,B)))
                 => ! [E] :
                      ( m2_pboole(E,A,k12_pralg_1(A,C))
                     => m1_subset_1(k4_card_3(k2_polynom1(A,k1_pzfmisc1(A,E),B,D)),u1_pre_topc(k6_pencil_1(A,C))) ) ) ) ) ) )).

fof(t17_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k6_pencil_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k6_pencil_1(A,B)))
                 => ( C != D
                   => ( r1_pencil_1(k6_pencil_1(A,B),C,D)
                    <=> ! [E] :
                          ( m1_pboole(E,A)
                         => ! [F] :
                              ( m1_pboole(F,A)
                             => ~ ( E = C
                                  & F = D
                                  & ! [G] :
                                      ( m1_subset_1(G,A)
                                     => ~ ( ! [H] :
                                              ( m1_subset_1(H,u1_struct_0(k2_pencil_1(A,B,G)))
                                             => ! [I] :
                                                  ( m1_subset_1(I,u1_struct_0(k2_pencil_1(A,B,G)))
                                                 => ( ( H = k1_funct_1(E,G)
                                                      & I = k1_funct_1(F,G) )
                                                   => ( H != I
                                                      & r1_pencil_1(k2_pencil_1(A,B,G),H,I) ) ) ) )
                                          & ! [H] :
                                              ( m1_subset_1(H,A)
                                             => ( H != G
                                               => k1_funct_1(E,H) = k1_funct_1(F,H) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( ~ v13_pencil_1(C)
                & v16_pencil_1(C,A)
                & m4_pboole(C,A,k12_pralg_1(A,B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,C))))
                 => ? [E] :
                      ( m1_pboole(E,A)
                      & r2_hidden(E,k4_card_3(C))
                      & k1_tarski(k2_polynom1(A,E,k3_pencil_1(A,C),D)) = k4_card_3(k2_polynom1(A,C,k3_pencil_1(A,C),k1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,C)),D))) ) ) ) ) ) )).

fof(t19_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_pboole(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( k1_funct_1(B,D) != k1_funct_1(C,D)
                   => k1_pencil_3(A,B,C) = k1_nat_1(k1_pencil_3(A,B,k2_polynom1(A,C,D,k1_funct_1(B,D))),np__1) ) ) ) ) ) )).

fof(d2_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_pencil_2(C,A,B)
             => ! [D] :
                  ( m1_pencil_2(D,A,B)
                 => ( r1_pencil_3(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k6_pencil_1(A,B)))
                       => ~ ( r2_hidden(E,C)
                            & ! [F] :
                                ( m1_subset_1(F,u1_struct_0(k6_pencil_1(A,B)))
                               => ~ ( r2_hidden(F,D)
                                    & r1_pencil_1(k6_pencil_1(A,B),E,F) ) ) ) ) ) ) ) ) ) )).

fof(t20_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_pencil_2(C,A,B)
             => ! [D] :
                  ( m1_pencil_2(D,A,B)
                 => ( r1_pencil_3(A,B,C,D)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                          & v1_pencil_2(E,k6_pencil_1(A,B),k6_pencil_1(A,B))
                          & m2_relset_1(E,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
                       => ! [F] :
                            ( m1_pencil_2(F,A,B)
                           => ! [G] :
                                ( m1_pencil_2(G,A,B)
                               => ( ( F = k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),E,C)
                                    & G = k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),E,D) )
                                 => r1_pencil_3(A,B,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t21_pencil_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_pencil_2(C,A,B)
             => ! [D] :
                  ( m1_pencil_2(D,A,B)
                 => ( r1_xboole_0(C,D)
                   => ( r1_pencil_3(A,B,C,D)
                    <=> ! [E] :
                          ( ( ~ v13_pencil_1(E)
                            & v16_pencil_1(E,A)
                            & m4_pboole(E,A,k12_pralg_1(A,B)) )
                         => ! [F] :
                              ( ( ~ v13_pencil_1(F)
                                & v16_pencil_1(F,A)
                                & m4_pboole(F,A,k12_pralg_1(A,B)) )
                             => ( ( C = k4_card_3(E)
                                  & D = k4_card_3(F) )
                               => ( k3_pencil_1(A,E) = k3_pencil_1(A,F)
                                  & ? [G] :
                                      ( m1_subset_1(G,A)
                                      & G != k3_pencil_1(A,E)
                                      & ! [H] :
                                          ( m1_subset_1(H,A)
                                         => ( H != G
                                           => k1_funct_1(E,H) = k1_funct_1(F,H) ) )
                                      & ! [H] :
                                          ( m1_subset_1(H,u1_struct_0(k2_pencil_1(A,B,G)))
                                         => ! [I] :
                                              ( m1_subset_1(I,u1_struct_0(k2_pencil_1(A,B,G)))
                                             => ( ( k1_funct_1(E,G) = k1_struct_0(k2_pencil_1(A,B,G),H)
                                                  & k1_funct_1(F,G) = k1_struct_0(k2_pencil_1(A,B,G),I) )
                                               => r1_pencil_1(k2_pencil_1(A,B,G),H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t22_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v9_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C)))
                   => ! [E] :
                        ( ( ~ v13_pencil_1(E)
                          & v16_pencil_1(E,A)
                          & m4_pboole(E,A,k12_pralg_1(A,B)) )
                       => ! [F] :
                            ( ( ~ v13_pencil_1(F)
                              & v16_pencil_1(F,A)
                              & m4_pboole(F,A,k12_pralg_1(A,B)) )
                           => ~ ( m1_pencil_2(k4_card_3(E),A,B)
                                & m1_pencil_2(k4_card_3(F),A,B)
                                & r6_pboole(A,E,k2_polynom1(A,F,C,k1_struct_0(k2_pencil_1(A,B,C),D)))
                                & ~ r2_hidden(D,k1_funct_1(F,C))
                                & ! [G] :
                                    ( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
                                   => ~ ( k1_funct_1(G,np__1) = k4_card_3(E)
                                        & k1_funct_1(G,k3_finseq_1(G)) = k4_card_3(F)
                                        & ! [H] :
                                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                                           => ( r2_hidden(H,k4_finseq_1(G))
                                             => m1_pencil_2(k1_funct_1(G,H),A,B) ) )
                                        & ! [H] :
                                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                                           => ( r1_xreal_0(np__1,H)
                                             => ( r1_xreal_0(k3_finseq_1(G),H)
                                                | ! [I] :
                                                    ( m1_pencil_2(I,A,B)
                                                   => ! [J] :
                                                        ( m1_pencil_2(J,A,B)
                                                       => ( ( I = k1_funct_1(G,H)
                                                            & J = k1_funct_1(G,k1_nat_1(H,np__1)) )
                                                         => ( r1_xboole_0(I,J)
                                                            & r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t23_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v9_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( m1_pencil_2(C,A,B)
               => ! [D] :
                    ( m1_pencil_2(D,A,B)
                   => ( r1_xboole_0(C,D)
                     => ! [E] :
                          ( ( ~ v13_pencil_1(E)
                            & v16_pencil_1(E,A)
                            & m4_pboole(E,A,k12_pralg_1(A,B)) )
                         => ! [F] :
                              ( ( ~ v13_pencil_1(F)
                                & v16_pencil_1(F,A)
                                & m4_pboole(F,A,k12_pralg_1(A,B)) )
                             => ( ( C = k4_card_3(E)
                                  & D = k4_card_3(F) )
                               => ( k3_pencil_1(A,E) = k3_pencil_1(A,F)
                                <=> ? [G] :
                                      ( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
                                      & k1_funct_1(G,np__1) = C
                                      & k1_funct_1(G,k3_finseq_1(G)) = D
                                      & ! [H] :
                                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                                         => ( r2_hidden(H,k4_finseq_1(G))
                                           => m1_pencil_2(k1_funct_1(G,H),A,B) ) )
                                      & ! [H] :
                                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                                         => ( r1_xreal_0(np__1,H)
                                           => ( r1_xreal_0(k3_finseq_1(G),H)
                                              | ! [I] :
                                                  ( m1_pencil_2(I,A,B)
                                                 => ! [J] :
                                                      ( m1_pencil_2(J,A,B)
                                                     => ( ( I = k1_funct_1(G,H)
                                                          & J = k1_funct_1(G,k1_nat_1(H,np__1)) )
                                                       => ( r1_xboole_0(I,J)
                                                          & r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t24_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( m1_pencil_2(D,A,B)
                   => ! [E] :
                        ( m1_pencil_2(E,A,B)
                       => ! [F] :
                            ( ( ~ v13_pencil_1(F)
                              & v16_pencil_1(F,A)
                              & m4_pboole(F,A,k12_pralg_1(A,B)) )
                           => ! [G] :
                                ( ( ~ v13_pencil_1(G)
                                  & v16_pencil_1(G,A)
                                  & m4_pboole(G,A,k12_pralg_1(A,B)) )
                               => ! [H] :
                                    ( ( ~ v13_pencil_1(H)
                                      & v16_pencil_1(H,A)
                                      & m4_pboole(H,A,k12_pralg_1(A,B)) )
                                   => ! [I] :
                                        ( ( ~ v13_pencil_1(I)
                                          & v16_pencil_1(I,A)
                                          & m4_pboole(I,A,k12_pralg_1(A,B)) )
                                       => ( ( D = k4_card_3(F)
                                            & E = k4_card_3(G)
                                            & k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,D) = k4_card_3(H)
                                            & k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,E) = k4_card_3(I)
                                            & k3_pencil_1(A,F) = k3_pencil_1(A,G) )
                                         => k3_pencil_1(A,H) = k3_pencil_1(A,I) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t25_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,A,A)
                    & v3_funct_2(D,A,A)
                    & m2_relset_1(D,A,A)
                    & ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,A)
                           => ( k8_funct_2(A,A,D,E) = F
                            <=> ! [G] :
                                  ( m1_pencil_2(G,A,B)
                                 => ! [H] :
                                      ( ( ~ v13_pencil_1(H)
                                        & v16_pencil_1(H,A)
                                        & m4_pboole(H,A,k12_pralg_1(A,B)) )
                                     => ! [I] :
                                          ( ( ~ v13_pencil_1(I)
                                            & v16_pencil_1(I,A)
                                            & m4_pboole(I,A,k12_pralg_1(A,B)) )
                                         => ( ( G = k4_card_3(H)
                                              & k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,G) = k4_card_3(I)
                                              & k3_pencil_1(A,H) = E )
                                           => k3_pencil_1(A,I) = F ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,A,A)
                      & v3_funct_2(D,A,A)
                      & m2_relset_1(D,A,A) )
                   => ( D = k2_pencil_3(A,B,C)
                    <=> ! [E] :
                          ( m1_subset_1(E,A)
                         => ! [F] :
                              ( m1_subset_1(F,A)
                             => ( k8_funct_2(A,A,D,E) = F
                              <=> ! [G] :
                                    ( m1_pencil_2(G,A,B)
                                   => ! [H] :
                                        ( ( ~ v13_pencil_1(H)
                                          & v16_pencil_1(H,A)
                                          & m4_pboole(H,A,k12_pralg_1(A,B)) )
                                       => ! [I] :
                                            ( ( ~ v13_pencil_1(I)
                                              & v16_pencil_1(I,A)
                                              & m4_pboole(I,A,k12_pralg_1(A,B)) )
                                           => ( ( G = k4_card_3(H)
                                                & k4_pre_topc(k6_pencil_1(A,B),k6_pencil_1(A,B),C,G) = k4_card_3(I)
                                                & k3_pencil_1(A,H) = E )
                                             => k3_pencil_1(A,I) = F ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( ( ~ v13_pencil_1(D)
                      & v16_pencil_1(D,A)
                      & m4_pboole(D,A,k12_pralg_1(A,B)) )
                   => ( m1_pencil_2(k4_card_3(D),A,B)
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D)))))
                            & m2_relset_1(E,u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))) )
                         => ( E = k3_pencil_3(A,B,C,D)
                          <=> ( v1_pencil_2(E,k2_pencil_1(A,B,k3_pencil_1(A,D)),k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))
                              & ! [F] :
                                  ( m1_pboole(F,A)
                                 => ( ( m1_subset_1(F,u1_struct_0(k6_pencil_1(A,B)))
                                      & r2_hidden(F,k4_card_3(D)) )
                                   => ! [G] :
                                        ( m1_pboole(G,A)
                                       => ( G = k1_funct_1(C,F)
                                         => k1_funct_1(G,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))) = k1_funct_1(E,k1_funct_1(F,k3_pencil_1(A,D))) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t26_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( m1_pencil_2(D,A,B)
                   => ! [E] :
                        ( m1_pencil_2(E,A,B)
                       => ( ( r1_xboole_0(D,E)
                            & r1_pencil_3(A,B,D,E) )
                         => ! [F] :
                              ( ( ~ v13_pencil_1(F)
                                & v16_pencil_1(F,A)
                                & m4_pboole(F,A,k12_pralg_1(A,B)) )
                             => ! [G] :
                                  ( ( ~ v13_pencil_1(G)
                                    & v16_pencil_1(G,A)
                                    & m4_pboole(G,A,k12_pralg_1(A,B)) )
                                 => ( ( k4_card_3(F) = D
                                      & k4_card_3(G) = E )
                                   => k3_pencil_3(A,B,C,F) = k3_pencil_3(A,B,C,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t27_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( ( ~ v13_pencil_1(D)
                      & v16_pencil_1(D,A)
                      & m4_pboole(D,A,k12_pralg_1(A,B)) )
                   => ! [E] :
                        ( ( ~ v13_pencil_1(E)
                          & v16_pencil_1(E,A)
                          & m4_pboole(E,A,k12_pralg_1(A,B)) )
                       => ( ( m1_pencil_2(k4_card_3(D),A,B)
                            & m1_pencil_2(k4_card_3(E),A,B)
                            & k3_pencil_1(A,D) = k3_pencil_1(A,E) )
                         => k3_pencil_3(A,B,C,D) = k3_pencil_3(A,B,C,E) ) ) ) ) ) ) ) )).

fof(d5_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D))))
                          & m2_relset_1(E,u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D)))) )
                       => ( E = k4_pencil_3(A,B,C,D)
                        <=> ! [F] :
                              ( ( ~ v13_pencil_1(F)
                                & v16_pencil_1(F,A)
                                & m4_pboole(F,A,k12_pralg_1(A,B)) )
                             => ( ( m1_pencil_2(k4_card_3(F),A,B)
                                  & k3_pencil_1(A,F) = D )
                               => E = k3_pencil_3(A,B,C,F) ) ) ) ) ) ) ) ) ) )).

fof(t28_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v15_pencil_1(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v10_pencil_1(k2_pencil_1(A,B,C)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
                  & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
                  & m2_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
               => ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,A,A)
                    & v3_funct_2(D,A,A)
                    & m2_relset_1(D,A,A)
                    & ? [E] :
                        ( v1_funcop_1(E)
                        & m1_pboole(E,A)
                        & ! [F] :
                            ( m1_subset_1(F,A)
                           => ( v1_funct_1(k1_funct_1(E,F))
                              & v1_funct_2(k1_funct_1(E,F),u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
                              & m2_relset_1(k1_funct_1(E,F),u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
                              & ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
                                    & m2_relset_1(G,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F)))) )
                                 => ( G = k1_funct_1(E,F)
                                   => v1_pencil_2(G,k2_pencil_1(A,B,F),k2_pencil_1(A,B,k8_funct_2(A,A,D,F))) ) )
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k6_pencil_1(A,B)))
                                 => ! [H] :
                                      ( m1_pboole(H,A)
                                     => ( H = G
                                       => ! [I] :
                                            ( m1_pboole(I,A)
                                           => ( I = k8_funct_2(u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)),C,G)
                                             => ! [J] :
                                                  ( ( v1_funct_1(J)
                                                    & v1_funct_2(J,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F))))
                                                    & m2_relset_1(J,u1_struct_0(k2_pencil_1(A,B,F)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,D,F)))) )
                                                 => ( J = k1_funct_1(E,F)
                                                   => k1_funct_1(I,k8_funct_2(A,A,D,F)) = k1_funct_1(J,k1_funct_1(H,F)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_pencil_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_pboole(B,A)
        & m1_pboole(C,A) )
     => m2_subset_1(k1_pencil_3(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k2_pencil_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
        & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
        & m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B))) )
     => ( v1_funct_1(k2_pencil_3(A,B,C))
        & v1_funct_2(k2_pencil_3(A,B,C),A,A)
        & v3_funct_2(k2_pencil_3(A,B,C),A,A)
        & m2_relset_1(k2_pencil_3(A,B,C),A,A) ) ) )).

fof(dt_k3_pencil_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
        & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
        & m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
        & ~ v13_pencil_1(D)
        & v16_pencil_1(D,A)
        & m4_pboole(D,A,k12_pralg_1(A,B)) )
     => ( v1_funct_1(k3_pencil_3(A,B,C,D))
        & v1_funct_2(k3_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D)))))
        & m2_relset_1(k3_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D))),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),k3_pencil_1(A,D))))) ) ) )).

fof(dt_k4_pencil_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v15_pencil_1(B)
        & m1_pboole(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
        & v1_pencil_2(C,k6_pencil_1(A,B),k6_pencil_1(A,B))
        & m1_relset_1(C,u1_struct_0(k6_pencil_1(A,B)),u1_struct_0(k6_pencil_1(A,B)))
        & m1_subset_1(D,A) )
     => ( v1_funct_1(k4_pencil_3(A,B,C,D))
        & v1_funct_2(k4_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D))))
        & m2_relset_1(k4_pencil_3(A,B,C,D),u1_struct_0(k2_pencil_1(A,B,D)),u1_struct_0(k2_pencil_1(A,B,k8_funct_2(A,A,k2_pencil_3(A,B,C),D)))) ) ) )).

fof(d1_pencil_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_pboole(C,A)
             => k1_pencil_3(A,B,C) = k1_card_1(a_3_0_pencil_3(A,B,C)) ) ) ) )).

fof(fraenkel_a_3_0_pencil_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & m1_pboole(C,B)
        & m1_pboole(D,B) )
     => ( r2_hidden(A,a_3_0_pencil_3(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & k1_funct_1(C,E) != k1_funct_1(D,E) ) ) ) )).
%------------------------------------------------------------------------------