SET007 Axioms: SET007+408.ax


%------------------------------------------------------------------------------
% File     : SET007+408 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Projective Planes
% 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    : projpl_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   50 (   0 unit)
%            Number of atoms       :  691 (  55 equality)
%            Maximal formula depth :   37 (  16 average)
%            Number of connectives :  728 (  87 ~  ;  15  |; 336  &)
%                                         (  15 <=>; 275 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   0 propositional; 1-5 arity)
%            Number of functors    :    9 (   0 constant; 1-5 arity)
%            Number of variables   :  278 (   0 singleton; 255 !;  23 ?)
%            Maximal term depth    :    3 (   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(d1_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ( r1_projpl_1(A,B,C,D)
                  <=> ( ~ r1_incsp_1(A,B,D)
                      & ~ r1_incsp_1(A,C,D) ) ) ) ) ) ) )).

fof(d2_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ( r2_projpl_1(A,B,C,D)
                  <=> ( r1_incsp_1(A,B,C)
                      & r1_incsp_1(A,B,D) ) ) ) ) ) ) )).

fof(d3_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ( r3_projpl_1(A,B,C,D,E)
                      <=> ( r1_incsp_1(A,B,C)
                          & r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,B,E) ) ) ) ) ) ) ) )).

fof(t1_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ( ( r3_incproj(A,B,C,E)
                                 => r3_incproj(A,C,B,E) )
                                & ( r4_incproj(A,B,C,E,D)
                                 => ( r4_incproj(A,B,D,E,C)
                                    & r4_incproj(A,C,B,E,D)
                                    & r4_incproj(A,C,D,E,B)
                                    & r4_incproj(A,D,B,E,C)
                                    & r4_incproj(A,D,C,E,B) ) )
                                & ( r2_projpl_1(A,B,E,F)
                                 => r2_projpl_1(A,B,F,E) )
                                & ( r3_projpl_1(A,B,E,F,G)
                                 => ( r3_projpl_1(A,B,E,G,F)
                                    & r3_projpl_1(A,B,F,E,G)
                                    & r3_projpl_1(A,B,F,G,E)
                                    & r3_projpl_1(A,B,G,E,F)
                                    & r3_projpl_1(A,B,G,F,E) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( v1_projpl_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_incsp_1(A))
           => ! [C] :
                ( m1_subset_1(C,u1_incsp_1(A))
               => ! [D] :
                    ( m1_subset_1(D,u2_incsp_1(A))
                   => ! [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                       => ~ ( r1_incsp_1(A,B,D)
                            & r1_incsp_1(A,C,D)
                            & r1_incsp_1(A,B,E)
                            & r1_incsp_1(A,C,E)
                            & B != C
                            & D != E ) ) ) ) ) ) ) )).

fof(t2_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( v1_projpl_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_incsp_1(A))
           => ! [C] :
                ( m1_subset_1(C,u1_incsp_1(A))
               => ! [D] :
                    ( m1_subset_1(D,u2_incsp_1(A))
                   => ! [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                       => ~ ( r3_incproj(A,B,C,D)
                            & r3_incproj(A,B,C,E)
                            & B != C
                            & D != E ) ) ) ) ) ) ) )).

fof(t3_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( v1_projpl_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_incsp_1(A))
           => ! [C] :
                ( m1_subset_1(C,u1_incsp_1(A))
               => ! [D] :
                    ( m1_subset_1(D,u2_incsp_1(A))
                   => ! [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                       => ~ ( r2_projpl_1(A,B,D,E)
                            & r2_projpl_1(A,C,D,E)
                            & B != C
                            & D != E ) ) ) ) ) ) ) )).

fof(t4_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( ( v1_incproj(A)
          & v2_incproj(A)
          & v3_incproj(A)
          & v4_incproj(A)
          & v5_incproj(A)
          & l1_incsp_1(A) )
      <=> ( v1_projpl_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u2_incsp_1(A))
                      & r3_incproj(A,B,C,D) ) ) )
          & ~ ! [B] :
                ( m1_subset_1(B,u1_incsp_1(A))
               => ! [C] :
                    ( m1_subset_1(C,u2_incsp_1(A))
                   => r1_incsp_1(A,B,C) ) )
          & ! [B] :
              ( m1_subset_1(B,u2_incsp_1(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                  & ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & ? [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                          & r1_incproj(C,D,E)
                          & r4_incproj(A,C,D,B,E) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ! [F] :
                              ( m1_subset_1(F,u1_incsp_1(A))
                             => ! [G] :
                                  ( m1_subset_1(G,u2_incsp_1(A))
                                 => ! [H] :
                                      ( m1_subset_1(H,u2_incsp_1(A))
                                     => ! [I] :
                                          ( m1_subset_1(I,u2_incsp_1(A))
                                         => ! [J] :
                                              ( m1_subset_1(J,u2_incsp_1(A))
                                             => ~ ( r4_incproj(A,B,C,G,F)
                                                  & r4_incproj(A,D,E,H,F)
                                                  & r3_incproj(A,B,D,I)
                                                  & r3_incproj(A,C,E,J)
                                                  & ~ r1_incsp_1(A,F,I)
                                                  & ~ r1_incsp_1(A,F,J)
                                                  & G != H
                                                  & ! [K] :
                                                      ( m1_subset_1(K,u1_incsp_1(A))
                                                     => ~ r2_projpl_1(A,K,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( r4_projpl_1(A,B,C,D)
                  <=> ? [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                        & r4_incproj(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t5_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( r4_projpl_1(A,B,C,D)
                  <=> ? [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                        & r1_incsp_1(A,B,E)
                        & r1_incsp_1(A,C,E)
                        & r1_incsp_1(A,D,E) ) ) ) ) ) ) )).

fof(t6_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( ~ r4_projpl_1(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u2_incsp_1(A))
                       => ~ ( r1_incsp_1(A,B,E)
                            & r1_incsp_1(A,C,E)
                            & r1_incsp_1(A,D,E) ) ) ) ) ) ) ) )).

fof(d6_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ( r5_projpl_1(A,B,C,D,E)
                      <=> ( ~ r4_projpl_1(A,B,C,D)
                          & ~ r4_projpl_1(A,C,D,E)
                          & ~ r4_projpl_1(A,D,E,B)
                          & ~ r4_projpl_1(A,E,B,C) ) ) ) ) ) ) ) )).

fof(t7_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ~ ( v1_incproj(A)
          & v2_incproj(A)
          & v3_incproj(A)
          & v4_incproj(A)
          & v5_incproj(A)
          & l1_incsp_1(A)
          & ! [B] :
              ( m1_subset_1(B,u2_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u2_incsp_1(A))
                 => B = C ) ) ) ) )).

fof(t8_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ~ ( v1_incproj(A)
                  & v2_incproj(A)
                  & v3_incproj(A)
                  & v4_incproj(A)
                  & v5_incproj(A)
                  & l1_incsp_1(A)
                  & r1_incsp_1(A,B,C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ~ ( r3_incproj(A,D,E,C)
                              & r1_incproj(B,D,E) ) ) ) ) ) ) ) )).

fof(t9_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ~ ( v1_incproj(A)
                      & v2_incproj(A)
                      & v3_incproj(A)
                      & v4_incproj(A)
                      & v5_incproj(A)
                      & l1_incsp_1(A)
                      & r1_incsp_1(A,B,C)
                      & C != D
                      & ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ~ ( r1_incsp_1(A,E,C)
                              & ~ r1_incsp_1(A,E,D)
                              & B != E ) ) ) ) ) ) ) )).

fof(t10_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ~ ( v1_projpl_1(A)
                          & r3_incproj(A,B,C,E)
                          & B != C
                          & ~ r1_incsp_1(A,D,E)
                          & r4_projpl_1(A,B,C,D) ) ) ) ) ) ) )).

fof(t11_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( r4_projpl_1(A,B,C,D)
                   => ( r4_projpl_1(A,B,D,C)
                      & r4_projpl_1(A,C,B,D)
                      & r4_projpl_1(A,C,D,B)
                      & r4_projpl_1(A,D,B,C)
                      & r4_projpl_1(A,D,C,B) ) ) ) ) ) ) )).

fof(t12_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( ~ r4_projpl_1(A,B,C,D)
                   => ( ~ r4_projpl_1(A,B,D,C)
                      & ~ r4_projpl_1(A,C,B,D)
                      & ~ r4_projpl_1(A,C,D,B)
                      & ~ r4_projpl_1(A,D,B,C)
                      & ~ r4_projpl_1(A,D,C,B) ) ) ) ) ) ) )).

fof(t13_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ( r5_projpl_1(A,B,C,D,E)
                       => ( r5_projpl_1(A,B,D,C,E)
                          & r5_projpl_1(A,C,B,D,E)
                          & r5_projpl_1(A,C,D,B,E)
                          & r5_projpl_1(A,D,B,C,E)
                          & r5_projpl_1(A,D,C,B,E)
                          & r5_projpl_1(A,B,C,E,D)
                          & r5_projpl_1(A,B,D,E,C)
                          & r5_projpl_1(A,C,B,E,D)
                          & r5_projpl_1(A,C,D,E,B)
                          & r5_projpl_1(A,D,B,E,C)
                          & r5_projpl_1(A,D,C,E,B)
                          & r5_projpl_1(A,B,E,C,D)
                          & r5_projpl_1(A,B,E,D,C)
                          & r5_projpl_1(A,C,E,B,D)
                          & r5_projpl_1(A,C,E,D,B)
                          & r5_projpl_1(A,D,E,B,C)
                          & r5_projpl_1(A,D,E,C,B)
                          & r5_projpl_1(A,E,B,C,D)
                          & r5_projpl_1(A,E,B,D,C)
                          & r5_projpl_1(A,E,C,B,D)
                          & r5_projpl_1(A,E,C,D,B)
                          & r5_projpl_1(A,E,D,B,C)
                          & r5_projpl_1(A,E,D,C,B) ) ) ) ) ) ) ) )).

fof(t14_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ( ( v1_projpl_1(A)
                                  & r3_incproj(A,B,C,F)
                                  & r3_incproj(A,D,E,G)
                                  & r1_projpl_1(A,B,C,G)
                                  & r1_projpl_1(A,D,E,F) )
                               => ( B = C
                                  | D = E
                                  | r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) ) ) ) )).

fof(t15_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ~ ( v1_incproj(A)
          & v2_incproj(A)
          & v3_incproj(A)
          & v4_incproj(A)
          & v5_incproj(A)
          & l1_incsp_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ~ r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) )).

fof(d7_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k4_zfmisc_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A)))
         => ( m1_projpl_1(B,A)
          <=> r5_projpl_1(A,k8_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k9_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k10_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k11_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B)) ) ) ) )).

fof(d8_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ( B != C
               => ! [D] :
                    ( m1_subset_1(D,u2_incsp_1(A))
                   => ( D = k1_projpl_1(A,B,C)
                    <=> r3_incproj(A,B,C,D) ) ) ) ) ) ) )).

fof(t16_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ( B != C
                   => ( r1_incsp_1(A,B,k1_projpl_1(A,B,C))
                      & r1_incsp_1(A,C,k1_projpl_1(A,B,C))
                      & k1_projpl_1(A,B,C) = k1_projpl_1(A,C,B)
                      & ( ( r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,C,D) )
                       => D = k1_projpl_1(A,B,C) ) ) ) ) ) ) ) )).

fof(t17_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ~ ( ~ ! [B] :
                ( m1_subset_1(B,u1_incsp_1(A))
               => ! [C] :
                    ( m1_subset_1(C,u1_incsp_1(A))
                   => ! [D] :
                        ( m1_subset_1(D,u1_incsp_1(A))
                       => r4_projpl_1(A,B,C,D) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u2_incsp_1(A))
                      & r3_incproj(A,B,C,D) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u2_incsp_1(A))
                 => r1_incsp_1(A,B,C) ) ) ) ) )).

fof(t18_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ~ ( ? [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
              & ? [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                  & ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & ? [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                          & r5_projpl_1(A,B,C,D,E) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => r4_projpl_1(A,B,C,D) ) ) ) ) ) )).

fof(t19_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ~ ( ~ r4_projpl_1(A,B,C,D)
                              & r3_incproj(A,B,C,E)
                              & r3_incproj(A,B,D,F)
                              & E = F ) ) ) ) ) ) ) )).

fof(t20_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ( ( r5_projpl_1(A,B,C,D,E)
                                      & r3_incproj(A,B,C,F)
                                      & r3_incproj(A,B,D,G)
                                      & r3_incproj(A,B,E,H) )
                                   => r1_incproj(F,G,H) ) ) ) ) ) ) ) ) ) )).

fof(t21_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ! [I] :
                                      ( m1_subset_1(I,u2_incsp_1(A))
                                     => ( ( v1_projpl_1(A)
                                          & r3_projpl_1(A,B,F,G,H)
                                          & r1_incproj(F,G,H)
                                          & r2_projpl_1(A,C,I,F)
                                          & r2_projpl_1(A,D,I,G)
                                          & r2_projpl_1(A,E,I,H) )
                                       => ( r1_incsp_1(A,B,I)
                                          | r1_incproj(C,D,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t22_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( ( v1_projpl_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u2_incsp_1(A))
                      & r3_incproj(A,B,C,D) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u2_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u2_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & r2_projpl_1(A,D,B,C) ) ) ) )
       => ( ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ~ r5_projpl_1(A,B,C,D,E) ) ) ) )
          | ! [B] :
              ( m1_subset_1(B,u2_incsp_1(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                  & ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & ? [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                          & r1_incproj(C,D,E)
                          & r4_incproj(A,C,D,B,E) ) ) ) ) ) ) ) )).

fof(t23_projpl_1,axiom,(
    ! [A] :
      ( l1_incsp_1(A)
     => ( ( v1_incproj(A)
          & v2_incproj(A)
          & v3_incproj(A)
          & v4_incproj(A)
          & v5_incproj(A)
          & v6_incproj(A)
          & l1_incsp_1(A) )
      <=> ( v1_projpl_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u2_incsp_1(A))
                      & r3_incproj(A,B,C,D) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u2_incsp_1(A))
             => ! [C] :
                  ( m1_subset_1(C,u2_incsp_1(A))
                 => ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & r2_projpl_1(A,D,B,C) ) ) )
          & ? [B] :
              ( m1_subset_1(B,u1_incsp_1(A))
              & ? [C] :
                  ( m1_subset_1(C,u1_incsp_1(A))
                  & ? [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                      & ? [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                          & r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) ) )).

fof(d9_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u2_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ( B != C
               => ! [D] :
                    ( m1_subset_1(D,u1_incsp_1(A))
                   => ( D = k2_projpl_1(A,B,C)
                    <=> r2_projpl_1(A,D,B,C) ) ) ) ) ) ) )).

fof(t24_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ( C != D
                   => ( r1_incsp_1(A,k2_projpl_1(A,C,D),C)
                      & r1_incsp_1(A,k2_projpl_1(A,C,D),D)
                      & k2_projpl_1(A,C,D) = k2_projpl_1(A,D,C)
                      & ( ( r1_incsp_1(A,B,C)
                          & r1_incsp_1(A,B,D) )
                       => B = k2_projpl_1(A,C,D) ) ) ) ) ) ) ) )).

fof(t25_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ( r1_incsp_1(A,B,D)
                       => ( D = E
                          | r1_incsp_1(A,C,D)
                          | B = k2_projpl_1(A,D,E)
                          | ( r1_incsp_1(A,k2_projpl_1(A,k1_projpl_1(A,C,B),E),E)
                            & ~ r1_incsp_1(A,k2_projpl_1(A,k1_projpl_1(A,C,B),E),D) ) ) ) ) ) ) ) ) )).

fof(t26_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( ~ r4_projpl_1(A,B,C,D)
                   => r1_incproj(B,C,D) ) ) ) ) ) )).

fof(t27_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ( r5_projpl_1(A,B,C,D,E)
                       => r2_incproj(B,C,D,E) ) ) ) ) ) ) )).

fof(t28_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ~ ( k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E)
                          & B != D
                          & C != E
                          & D != E
                          & k1_projpl_1(A,B,D) != k1_projpl_1(A,D,E) ) ) ) ) ) ) )).

fof(t29_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ~ ( k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E)
                          & B != D
                          & C != E
                          & D != E
                          & ~ r1_incsp_1(A,B,k1_projpl_1(A,D,E)) ) ) ) ) ) ) )).

fof(t30_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ( ( r1_incsp_1(A,C,E)
                          & r1_incsp_1(A,D,E) )
                       => ( C = D
                          | r1_incsp_1(A,B,E)
                          | ( k1_projpl_1(A,C,B) != k1_projpl_1(A,D,B)
                            & k1_projpl_1(A,B,C) != k1_projpl_1(A,B,D) ) ) ) ) ) ) ) ) )).

fof(t31_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ( ( r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,B,E) )
                       => ( D = E
                          | r1_incsp_1(A,B,C)
                          | ( k2_projpl_1(A,C,D) != k2_projpl_1(A,C,E)
                            & k2_projpl_1(A,D,C) != k2_projpl_1(A,E,C) ) ) ) ) ) ) ) ) )).

fof(t32_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ( ( r1_incsp_1(A,D,k1_projpl_1(A,B,C))
                          & r1_incsp_1(A,D,k1_projpl_1(A,B,E)) )
                       => ( D = B
                          | E = B
                          | B = C
                          | r1_incsp_1(A,E,k1_projpl_1(A,B,C)) ) ) ) ) ) ) ) )).

fof(t33_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ( r1_incsp_1(A,D,k1_projpl_1(A,B,C))
                   => ( B = D
                      | r1_incsp_1(A,C,k1_projpl_1(A,B,D)) ) ) ) ) ) ) )).

fof(t34_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ~ ( D != E
                              & r1_incsp_1(A,D,F)
                              & r1_incsp_1(A,E,F)
                              & ~ r1_incsp_1(A,B,F)
                              & ~ r1_incsp_1(A,C,F)
                              & k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E) ) ) ) ) ) ) ) )).

fof(t35_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ~ ( r1_incsp_1(A,B,k1_projpl_1(A,C,D))
                      & B != D
                      & C != D
                      & ~ r1_incsp_1(A,C,k1_projpl_1(A,D,B)) ) ) ) ) ) )).

fof(t36_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ~ ( r1_incsp_1(A,B,k1_projpl_1(A,C,D))
                      & C != B
                      & C != D
                      & ~ r1_incsp_1(A,D,k1_projpl_1(A,C,B)) ) ) ) ) ) )).

fof(t37_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ~ ( r1_incsp_1(A,C,H)
                                      & r1_incsp_1(A,D,H)
                                      & r1_incsp_1(A,B,G)
                                      & ~ r1_incsp_1(A,B,H)
                                      & C != D
                                      & E != B
                                      & F != B
                                      & r1_incsp_1(A,E,k1_projpl_1(A,B,C))
                                      & r1_incsp_1(A,F,k1_projpl_1(A,B,D))
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_incsp_1(A))
                                         => ~ ( r1_incsp_1(A,I,k1_projpl_1(A,E,F))
                                              & r1_incsp_1(A,I,G)
                                              & I != B ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_projpl_1(B,A)
         => m1_subset_1(B,k4_zfmisc_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A))) ) ) )).

fof(existence_m1_projpl_1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ? [B] : m1_projpl_1(B,A) ) )).

fof(dt_k1_projpl_1,axiom,(
    ! [A,B,C] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A)
        & m1_subset_1(B,u1_incsp_1(A))
        & m1_subset_1(C,u1_incsp_1(A)) )
     => m1_subset_1(k1_projpl_1(A,B,C),u2_incsp_1(A)) ) )).

fof(dt_k2_projpl_1,axiom,(
    ! [A,B,C] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A)
        & m1_subset_1(B,u2_incsp_1(A))
        & m1_subset_1(C,u2_incsp_1(A)) )
     => m1_subset_1(k2_projpl_1(A,B,C),u1_incsp_1(A)) ) )).
%------------------------------------------------------------------------------