SET007 Axioms: SET007+275.ax


%------------------------------------------------------------------------------
% File     : SET007+275 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Incidence Projective Space (a reduction theorem in a plane)
% 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    : projred1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   3 unit)
%            Number of atoms       :  502 (  28 equality)
%            Maximal formula depth :   59 (  18 average)
%            Number of connectives :  532 (  60 ~  ;  10  |; 321  &)
%                                         (   3 <=>; 138 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-5 arity)
%            Number of functors    :    8 (   0 constant; 1-4 arity)
%            Number of variables   :  177 (   0 singleton; 131 !;  46 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_projred1,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,u2_incsp_1(A))
         => ~ ! [C] :
                ( m1_subset_1(C,u1_incsp_1(A))
               => r1_incsp_1(A,C,B) ) ) ) )).

fof(t2_projred1,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,u2_incsp_1(A))
               => r1_incsp_1(A,B,C) ) ) ) )).

fof(t3_projred1,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,u2_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ~ ( B != C
                  & ! [D] :
                      ( m1_subset_1(D,u1_incsp_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_incsp_1(A))
                         => ~ ( r1_incsp_1(A,D,B)
                              & ~ r1_incsp_1(A,D,C)
                              & r1_incsp_1(A,E,C)
                              & ~ r1_incsp_1(A,E,B) ) ) ) ) ) ) ) )).

fof(t4_projred1,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))
                     => ! [E] :
                          ( m1_subset_1(E,u2_incsp_1(A))
                         => ~ ( r1_incsp_1(A,B,D)
                              & ~ r1_incsp_1(A,B,E)
                              & r1_incsp_1(A,C,E)
                              & ~ r1_incsp_1(A,C,D) ) ) ) ) ) ) ) )).

fof(t5_projred1,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,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,C)
                      & r1_incsp_1(A,B,D)
                      & r1_incsp_1(A,B,E)
                      & C != D
                      & D != E
                      & E != C ) ) ) ) ) )).

fof(t6_projred1,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,u2_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ? [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                  & ~ r1_incsp_1(A,D,B)
                  & ~ r1_incsp_1(A,D,C) ) ) ) ) )).

fof(t7_projred1,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,u2_incsp_1(A))
         => ? [C] :
              ( m1_subset_1(C,u1_incsp_1(A))
              & r1_incsp_1(A,C,B) ) ) ) )).

fof(t8_projred1,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))
                 => ? [E] :
                      ( m1_subset_1(E,u1_incsp_1(A))
                      & r1_incsp_1(A,E,D)
                      & E != B
                      & E != C ) ) ) ) ) )).

fof(t9_projred1,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))
                  & ~ r1_incsp_1(A,B,D)
                  & ~ r1_incsp_1(A,C,D) ) ) ) ) )).

fof(t10_projred1,axiom,(
    $true )).

fof(t11_projred1,axiom,(
    $true )).

fof(t12_projred1,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))
                     => ! [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))
                                     => ~ ( r1_incsp_1(A,B,F)
                                          & r1_incsp_1(A,B,G)
                                          & F != G
                                          & r1_incsp_1(A,C,F)
                                          & B != C
                                          & r1_incsp_1(A,D,G)
                                          & r1_incsp_1(A,E,G)
                                          & D != E
                                          & r1_incsp_1(A,C,H)
                                          & r1_incsp_1(A,D,H)
                                          & r1_incsp_1(A,C,I)
                                          & r1_incsp_1(A,E,I)
                                          & H = I ) ) ) ) ) ) ) ) ) ) )).

fof(t13_projred1,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,u2_incsp_1(A))
                     => ( 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) ) ) ) ) ) ) ) )).

fof(t14_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v10_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,u1_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_incsp_1(A))
                                 => ! [I] :
                                      ( m1_subset_1(I,u1_incsp_1(A))
                                     => ! [J] :
                                          ( m1_subset_1(J,u1_incsp_1(A))
                                         => ! [K] :
                                              ( m1_subset_1(K,u1_incsp_1(A))
                                             => ! [L] :
                                                  ( m1_subset_1(L,u2_incsp_1(A))
                                                 => ! [M] :
                                                      ( m1_subset_1(M,u2_incsp_1(A))
                                                     => ! [N] :
                                                          ( m1_subset_1(N,u2_incsp_1(A))
                                                         => ! [O] :
                                                              ( m1_subset_1(O,u2_incsp_1(A))
                                                             => ! [P] :
                                                                  ( m1_subset_1(P,u2_incsp_1(A))
                                                                 => ! [Q] :
                                                                      ( m1_subset_1(Q,u2_incsp_1(A))
                                                                     => ! [R] :
                                                                          ( m1_subset_1(R,u2_incsp_1(A))
                                                                         => ! [S] :
                                                                              ( m1_subset_1(S,u2_incsp_1(A))
                                                                             => ! [T] :
                                                                                  ( m1_subset_1(T,u2_incsp_1(A))
                                                                                 => ~ ( r4_incproj(A,B,C,L,D)
                                                                                      & r4_incproj(A,B,F,M,E)
                                                                                      & r4_incproj(A,B,H,N,G)
                                                                                      & r4_incproj(A,H,F,O,K)
                                                                                      & r4_incproj(A,H,I,P,D)
                                                                                      & r4_incproj(A,F,J,Q,D)
                                                                                      & r4_incproj(A,K,E,R,G)
                                                                                      & r4_incproj(A,C,I,S,G)
                                                                                      & r4_incproj(A,C,J,T,E)
                                                                                      & r1_incproj(L,M,N)
                                                                                      & B != H
                                                                                      & B != C
                                                                                      & B != E
                                                                                      & F != E
                                                                                      & ! [U] :
                                                                                          ( m1_subset_1(U,u2_incsp_1(A))
                                                                                         => ~ r4_incproj(A,I,J,U,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_projred1,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,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))
                        & ? [F] :
                            ( m1_subset_1(F,u1_incsp_1(A))
                            & r1_incsp_1(A,C,B)
                            & r1_incsp_1(A,D,B)
                            & r1_incsp_1(A,E,B)
                            & r1_incsp_1(A,F,B)
                            & r2_incproj(C,D,E,F) ) ) ) ) )
       => ! [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))
                        & ? [F] :
                            ( m1_subset_1(F,u1_incsp_1(A))
                            & r1_incsp_1(A,C,B)
                            & r1_incsp_1(A,D,B)
                            & r1_incsp_1(A,E,B)
                            & r1_incsp_1(A,F,B)
                            & r2_incproj(C,D,E,F) ) ) ) ) ) ) ) )).

fof(t16_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v9_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,u1_incsp_1(A))
                              & ? [H] :
                                  ( m1_subset_1(H,u1_incsp_1(A))
                                  & ? [I] :
                                      ( m1_subset_1(I,u2_incsp_1(A))
                                      & ? [J] :
                                          ( m1_subset_1(J,u2_incsp_1(A))
                                          & ? [K] :
                                              ( m1_subset_1(K,u2_incsp_1(A))
                                              & ? [L] :
                                                  ( m1_subset_1(L,u2_incsp_1(A))
                                                  & ? [M] :
                                                      ( m1_subset_1(M,u2_incsp_1(A))
                                                      & ? [N] :
                                                          ( m1_subset_1(N,u2_incsp_1(A))
                                                          & ? [O] :
                                                              ( m1_subset_1(O,u2_incsp_1(A))
                                                              & ? [P] :
                                                                  ( m1_subset_1(P,u2_incsp_1(A))
                                                                  & ~ r1_incsp_1(A,C,M)
                                                                  & ~ r1_incsp_1(A,D,M)
                                                                  & ~ r1_incsp_1(A,B,L)
                                                                  & ~ r1_incsp_1(A,E,L)
                                                                  & ~ r1_incsp_1(A,B,N)
                                                                  & ~ r1_incsp_1(A,D,N)
                                                                  & ~ r1_incsp_1(A,C,O)
                                                                  & ~ r1_incsp_1(A,E,O)
                                                                  & r4_incproj(A,F,B,M,E)
                                                                  & r4_incproj(A,F,C,L,D)
                                                                  & r4_incproj(A,G,C,N,E)
                                                                  & r4_incproj(A,G,B,O,D)
                                                                  & r4_incproj(A,H,B,I,C)
                                                                  & r4_incproj(A,H,D,J,E)
                                                                  & r3_incproj(A,F,G,K)
                                                                  & ~ r1_incsp_1(A,H,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v9_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))
                      & ? [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                          & r1_incsp_1(A,B,C)
                          & r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,B,E)
                          & r1_incsp_1(A,B,F)
                          & r2_incproj(C,D,E,F) ) ) ) ) ) ) )).

fof(t18_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v9_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))
                          & r1_incsp_1(A,B,F)
                          & r1_incsp_1(A,C,F)
                          & r1_incsp_1(A,D,F)
                          & r1_incsp_1(A,E,F)
                          & r2_incproj(B,C,D,E) ) ) ) ) ) ) )).

fof(t19_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v9_incproj(A)
        & l1_incsp_1(A) )
     => ! [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))
                      & ? [F] :
                          ( m1_subset_1(F,u1_incsp_1(A))
                          & r1_incsp_1(A,C,B)
                          & r1_incsp_1(A,D,B)
                          & r1_incsp_1(A,E,B)
                          & r1_incsp_1(A,F,B)
                          & r2_incproj(C,D,E,F) ) ) ) ) ) ) )).

fof(d1_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_incproj(A)
        & l1_incsp_1(A) )
     => ! [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))
                 => ~ ( ~ r1_incsp_1(A,D,B)
                      & ~ r1_incsp_1(A,D,C)
                      & ~ ! [E] :
                            ( ( v1_funct_1(E)
                              & m2_relset_1(E,u1_incsp_1(A),u1_incsp_1(A)) )
                           => ( E = k1_projred1(A,B,C,D)
                            <=> ( r1_tarski(k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),E),u1_incsp_1(A))
                                & ! [F] :
                                    ( m1_subset_1(F,u1_incsp_1(A))
                                   => ( r2_hidden(F,k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),E))
                                    <=> r1_incsp_1(A,F,B) ) )
                                & ! [F] :
                                    ( m1_subset_1(F,u1_incsp_1(A))
                                   => ! [G] :
                                        ( m1_subset_1(G,u1_incsp_1(A))
                                       => ( ( r1_incsp_1(A,F,B)
                                            & r1_incsp_1(A,G,C) )
                                         => ( k1_funct_1(E,F) = G
                                          <=> ? [H] :
                                                ( m1_subset_1(H,u2_incsp_1(A))
                                                & r1_incsp_1(A,D,H)
                                                & r1_incsp_1(A,F,H)
                                                & r1_incsp_1(A,G,H) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_projred1,axiom,(
    $true )).

fof(t21_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] :
              ( m1_subset_1(C,u2_incsp_1(A))
             => ( ~ r1_incsp_1(A,B,C)
               => ! [D] :
                    ( m1_subset_1(D,u1_incsp_1(A))
                   => ( r1_incsp_1(A,D,C)
                     => k1_funct_1(k1_projred1(A,C,C,B),D) = D ) ) ) ) ) ) )).

fof(t22_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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,C,D)
                       => ( r1_incsp_1(A,B,D)
                          | r1_incsp_1(A,B,E)
                          | m1_subset_1(k1_funct_1(k1_projred1(A,D,E,B),C),u1_incsp_1(A)) ) ) ) ) ) ) ) )).

fof(t23_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ( ( r1_incsp_1(A,C,E)
                              & D = k1_funct_1(k1_projred1(A,E,F,B),C) )
                           => ( r1_incsp_1(A,B,E)
                              | r1_incsp_1(A,B,F)
                              | r1_incsp_1(A,D,F) ) ) ) ) ) ) ) ) )).

fof(t24_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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))
                     => ( r2_hidden(C,k2_relat_1(k1_projred1(A,D,E,B)))
                       => ( r1_incsp_1(A,B,D)
                          | r1_incsp_1(A,B,E)
                          | r1_incsp_1(A,C,E) ) ) ) ) ) ) ) )).

fof(t25_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ~ ( ~ r1_incsp_1(A,B,D)
                              & ~ r1_incsp_1(A,B,E)
                              & ~ r1_incsp_1(A,C,E)
                              & ~ r1_incsp_1(A,C,F)
                              & ~ ( k1_relat_1(k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C))) = k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),k1_projred1(A,D,E,B))
                                  & k2_relat_1(k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C))) = k2_relat_1(k1_projred1(A,E,F,C)) ) ) ) ) ) ) ) ) )).

fof(t26_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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,u1_incsp_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u1_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_incsp_1(A))
                                 => ( ( r1_incsp_1(A,E,C)
                                      & r1_incsp_1(A,F,C)
                                      & k1_funct_1(k1_projred1(A,C,D,B),E) = G
                                      & k1_funct_1(k1_projred1(A,C,D,B),F) = H
                                      & G = H )
                                   => ( r1_incsp_1(A,B,C)
                                      | r1_incsp_1(A,B,D)
                                      | E = F ) ) ) ) ) ) ) ) ) ) )).

fof(t27_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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,C,D)
                          & r1_incsp_1(A,C,E) )
                       => ( r1_incsp_1(A,B,D)
                          | r1_incsp_1(A,B,E)
                          | k1_funct_1(k1_projred1(A,D,E,B),C) = C ) ) ) ) ) ) ) )).

fof(t28_projred1,axiom,(
    ! [A] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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))
                     => ! [F] :
                          ( m1_subset_1(F,u2_incsp_1(A))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ~ ( ~ r1_incsp_1(A,B,E)
                                  & ~ r1_incsp_1(A,B,F)
                                  & ~ r1_incsp_1(A,C,F)
                                  & ~ r1_incsp_1(A,C,G)
                                  & r1_incsp_1(A,D,E)
                                  & r1_incsp_1(A,D,F)
                                  & r1_incsp_1(A,D,G)
                                  & E != G
                                  & ! [H] :
                                      ( m1_subset_1(H,u1_incsp_1(A))
                                     => ~ ( ~ r1_incsp_1(A,H,E)
                                          & ~ r1_incsp_1(A,H,G)
                                          & k5_relat_1(k1_projred1(A,E,F,B),k1_projred1(A,F,G,C)) = k1_projred1(A,E,G,H) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_projred1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & v10_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(D,u1_incsp_1(A)) )
     => ( v1_funct_1(k1_projred1(A,B,C,D))
        & m2_relset_1(k1_projred1(A,B,C,D),u1_incsp_1(A),u1_incsp_1(A)) ) ) )).
%------------------------------------------------------------------------------