SET007 Axioms: SET007+283.ax


%------------------------------------------------------------------------------
% File     : SET007+283 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Projections in Projective Planes - Part II
% 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    : projred2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   32 (   1 unit)
%            Number of atoms       :  716 (  54 equality)
%            Maximal formula depth :   52 (  23 average)
%            Number of connectives :  787 ( 103 ~  ;  66  |; 389  &)
%                                         (   4 <=>; 225 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-5 arity)
%            Number of functors    :   12 (   0 constant; 1-4 arity)
%            Number of variables   :  226 (   0 singleton; 219 !;   7 ?)
%            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_projred2,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,u2_incsp_1(A))
                 => ( r1_projred2(A,B,C,D)
                  <=> ? [E] :
                        ( m1_subset_1(E,u1_incsp_1(A))
                        & r1_incsp_1(A,E,B)
                        & r1_incsp_1(A,E,C)
                        & r1_incsp_1(A,E,D) ) ) ) ) ) ) )).

fof(d3_projred2,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] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,u1_incsp_1(A),u1_incsp_1(A)) )
         => ( m1_projred2(B,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)
                        & B = k1_projred1(A,D,E,C) ) ) ) ) ) ) )).

fof(t1_projred2,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,u2_incsp_1(A))
                 => ( ~ ( B != C
                        & C != D
                        & D != B )
                   => r1_projred2(A,B,C,D) ) ) ) ) ) )).

fof(t2_projred2,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,u2_incsp_1(A))
                 => ( r1_projred2(A,B,C,D)
                   => ( r1_projred2(A,B,D,C)
                      & r1_projred2(A,C,B,D)
                      & r1_projred2(A,C,D,B)
                      & r1_projred2(A,D,B,C)
                      & r1_projred2(A,D,C,B) ) ) ) ) ) ) )).

fof(t3_projred2,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,B,D)
                          & ~ r1_incsp_1(A,B,E)
                          & r1_incsp_1(A,C,E)
                          & ! [F] :
                              ( m1_subset_1(F,u1_incsp_1(A))
                             => ~ ( r1_incsp_1(A,F,D)
                                  & k1_funct_1(k1_projred1(A,D,E,B),F) = C ) ) ) ) ) ) ) ) )).

fof(t4_projred2,axiom,(
    $true )).

fof(t5_projred2,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))
                 => ~ ( ~ r1_incsp_1(A,B,C)
                      & ~ r1_incsp_1(A,B,D)
                      & k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),k1_projred1(A,C,D,B)) != k1_projred2(A,C) ) ) ) ) ) )).

fof(t6_projred2,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))
                 => ~ ( ~ r1_incsp_1(A,B,C)
                      & ~ r1_incsp_1(A,B,D)
                      & k2_relat_1(k1_projred1(A,C,D,B)) != k1_projred2(A,D) ) ) ) ) ) )).

fof(t7_projred2,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] :
              ( r2_hidden(C,k1_projred2(A,B))
            <=> ? [D] :
                  ( m1_subset_1(D,u1_incsp_1(A))
                  & C = D
                  & r1_incsp_1(A,D,B) ) ) ) ) )).

fof(t8_projred2,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))
                 => ~ ( ~ r1_incsp_1(A,B,C)
                      & ~ r1_incsp_1(A,B,D)
                      & ~ v2_funct_1(k1_projred1(A,C,D,B)) ) ) ) ) ) )).

fof(t9_projred2,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))
                 => ~ ( ~ r1_incsp_1(A,B,C)
                      & ~ r1_incsp_1(A,B,D)
                      & k2_funct_1(k1_projred1(A,C,D,B)) != k1_projred1(A,D,C,B) ) ) ) ) ) )).

fof(t10_projred2,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_projred2(B,A)
         => m1_projred2(k2_funct_1(B),A) ) ) )).

fof(t11_projred2,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)
               => k1_projred1(A,C,C,B) = k6_relat_1(k1_projred2(A,C)) ) ) ) ) )).

fof(t12_projred2,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))
         => m1_projred2(k6_relat_1(k1_projred2(A,B)),A) ) ) )).

fof(t13_projred2,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,u2_incsp_1(A))
                     => ~ ( ~ r1_incsp_1(A,B,C)
                          & ~ r1_incsp_1(A,B,D)
                          & ~ r1_incsp_1(A,B,E)
                          & k5_relat_1(k1_projred1(A,C,E,B),k1_projred1(A,E,D,B)) != k1_projred1(A,C,D,B) ) ) ) ) ) ) )).

fof(t14_projred2,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)
                              & r1_projred2(A,D,E,F)
                              & D != F
                              & ! [G] :
                                  ( m1_subset_1(G,u1_incsp_1(A))
                                 => ~ ( ~ r1_incsp_1(A,G,D)
                                      & ~ r1_incsp_1(A,G,F)
                                      & k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C)) = k1_projred1(A,D,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t15_projred2,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,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,D,I)
                                                                      & r1_incsp_1(A,D,K)
                                                                      & r1_incsp_1(A,D,L)
                                                                      & r1_incsp_1(A,B,M)
                                                                      & r1_incsp_1(A,C,M)
                                                                      & r1_incsp_1(A,F,K)
                                                                      & r1_incsp_1(A,F,J)
                                                                      & r1_incsp_1(A,B,N)
                                                                      & r1_incsp_1(A,F,N)
                                                                      & r1_incsp_1(A,G,I)
                                                                      & r1_incsp_1(A,G,N)
                                                                      & r1_incsp_1(A,E,M)
                                                                      & r1_incsp_1(A,E,O)
                                                                      & r1_incsp_1(A,G,O)
                                                                      & r1_incsp_1(A,H,O)
                                                                      & r1_incsp_1(A,F,P)
                                                                      & r1_incsp_1(A,C,P)
                                                                      & r1_incsp_1(A,H,P)
                                                                      & r1_incsp_1(A,H,L) )
                                                                   => ( r1_incsp_1(A,B,I)
                                                                      | r1_incsp_1(A,C,J)
                                                                      | r1_incsp_1(A,B,K)
                                                                      | r1_incsp_1(A,C,K)
                                                                      | r1_projred2(A,I,J,K)
                                                                      | r1_incsp_1(A,C,L)
                                                                      | I = L
                                                                      | B = C
                                                                      | C = E
                                                                      | r1_projred2(A,J,K,M)
                                                                      | L = K
                                                                      | E = B
                                                                      | r1_incsp_1(A,E,I)
                                                                      | r1_incsp_1(A,E,L)
                                                                      | k5_relat_1(k1_projred1(A,I,K,B),k1_projred1(A,K,J,C)) = k5_relat_1(k1_projred1(A,I,L,E),k1_projred1(A,L,J,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t16_projred2,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,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,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))
                                                                 => ! [Q] :
                                                                      ( m1_subset_1(Q,u2_incsp_1(A))
                                                                     => ! [R] :
                                                                          ( m1_subset_1(R,u2_incsp_1(A))
                                                                         => ( ( r3_incproj(A,E,F,K)
                                                                              & r4_incproj(A,F,G,M,H)
                                                                              & r4_incproj(A,E,H,L,I)
                                                                              & r4_incproj(A,B,C,O,H)
                                                                              & r3_incproj(A,E,J,N)
                                                                              & r4_incproj(A,B,F,P,I)
                                                                              & r4_incproj(A,C,I,Q,J)
                                                                              & r4_incproj(A,F,J,R,D)
                                                                              & r1_incsp_1(A,D,O) )
                                                                           => ( r1_incsp_1(A,B,K)
                                                                              | r1_incsp_1(A,B,L)
                                                                              | r1_incsp_1(A,C,M)
                                                                              | r1_incsp_1(A,C,L)
                                                                              | r1_incsp_1(A,C,N)
                                                                              | r1_projred2(A,K,M,L)
                                                                              | B = C
                                                                              | C = D
                                                                              | K = N
                                                                              | k5_relat_1(k1_projred1(A,K,L,B),k1_projred1(A,L,M,C)) = k5_relat_1(k1_projred1(A,K,N,D),k1_projred1(A,N,M,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_projred2,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,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))
                                                                 => ( ( r3_incproj(A,D,E,I)
                                                                      & r1_incsp_1(A,F,K)
                                                                      & r3_incproj(A,D,F,J)
                                                                      & r4_incproj(A,B,C,M,G)
                                                                      & r3_incproj(A,D,H,L)
                                                                      & r4_incproj(A,B,F,N,E)
                                                                      & r4_incproj(A,G,E,O,H)
                                                                      & r4_incproj(A,C,F,P,H) )
                                                                   => ( r1_incsp_1(A,B,I)
                                                                      | r1_incsp_1(A,B,J)
                                                                      | r1_incsp_1(A,C,K)
                                                                      | r1_incsp_1(A,C,J)
                                                                      | r1_incsp_1(A,C,L)
                                                                      | r1_projred2(A,I,K,J)
                                                                      | r1_projred2(A,K,J,M)
                                                                      | I = L
                                                                      | L = J
                                                                      | B = C
                                                                      | ( G != B
                                                                        & G != C
                                                                        & ~ r1_incsp_1(A,G,I)
                                                                        & ~ r1_incsp_1(A,G,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_projred2,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,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,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))
                                                                 => ! [Q] :
                                                                      ( m1_subset_1(Q,u2_incsp_1(A))
                                                                     => ! [R] :
                                                                          ( m1_subset_1(R,u2_incsp_1(A))
                                                                         => ( ( r3_incproj(A,D,E,K)
                                                                              & r4_incproj(A,E,F,M,G)
                                                                              & r4_incproj(A,D,G,L,H)
                                                                              & r4_incproj(A,B,C,O,G)
                                                                              & r3_incproj(A,D,I,N)
                                                                              & r4_incproj(A,B,E,P,H)
                                                                              & r4_incproj(A,C,H,Q,I)
                                                                              & r4_incproj(A,E,I,R,J)
                                                                              & r1_incsp_1(A,J,O) )
                                                                           => ( r1_incsp_1(A,B,K)
                                                                              | r1_incsp_1(A,B,L)
                                                                              | r1_incsp_1(A,C,M)
                                                                              | r1_incsp_1(A,C,L)
                                                                              | r1_incsp_1(A,C,N)
                                                                              | r1_projred2(A,K,M,L)
                                                                              | B = C
                                                                              | K = N
                                                                              | ( ~ r1_incsp_1(A,J,K)
                                                                                & ~ r1_incsp_1(A,J,N)
                                                                                & C != J ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t19_projred2,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,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))
                                                                 => ( ( r3_incproj(A,E,F,I)
                                                                      & r1_incsp_1(A,G,K)
                                                                      & r3_incproj(A,E,G,J)
                                                                      & r4_incproj(A,B,C,L,D)
                                                                      & r3_incproj(A,E,H,M)
                                                                      & r4_incproj(A,B,G,N,F)
                                                                      & r4_incproj(A,D,F,O,H)
                                                                      & r4_incproj(A,C,G,P,H) )
                                                                   => ( r1_incsp_1(A,B,I)
                                                                      | r1_incsp_1(A,B,J)
                                                                      | r1_incsp_1(A,C,K)
                                                                      | r1_incsp_1(A,C,J)
                                                                      | r1_incsp_1(A,D,I)
                                                                      | r1_projred2(A,I,K,J)
                                                                      | r1_projred2(A,K,J,L)
                                                                      | B = C
                                                                      | C = D
                                                                      | D = B
                                                                      | ( M != I
                                                                        & M != J
                                                                        & ~ r1_incsp_1(A,D,M)
                                                                        & ~ r1_incsp_1(A,C,M) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_projred2,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,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,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))
                                                                 => ! [Q] :
                                                                      ( m1_subset_1(Q,u2_incsp_1(A))
                                                                     => ! [R] :
                                                                          ( m1_subset_1(R,u2_incsp_1(A))
                                                                         => ( ( r3_incproj(A,E,F,K)
                                                                              & r4_incproj(A,F,G,M,H)
                                                                              & r4_incproj(A,E,H,L,I)
                                                                              & r4_incproj(A,B,C,N,H)
                                                                              & r3_incproj(A,E,J,O)
                                                                              & r4_incproj(A,B,F,P,I)
                                                                              & r4_incproj(A,C,I,Q,J)
                                                                              & r4_incproj(A,F,J,R,D)
                                                                              & r1_incsp_1(A,D,N) )
                                                                           => ( r1_incsp_1(A,B,K)
                                                                              | r1_incsp_1(A,B,L)
                                                                              | r1_incsp_1(A,C,M)
                                                                              | r1_incsp_1(A,C,L)
                                                                              | r1_incsp_1(A,D,K)
                                                                              | r1_projred2(A,K,M,L)
                                                                              | B = C
                                                                              | C = D
                                                                              | ( ~ r1_incsp_1(A,C,O)
                                                                                & ~ r1_incsp_1(A,D,O)
                                                                                & K != O ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t21_projred2,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))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ~ ( ~ r1_incsp_1(A,B,D)
                                      & ~ r1_incsp_1(A,C,E)
                                      & ~ r1_incsp_1(A,B,F)
                                      & ~ r1_incsp_1(A,C,F)
                                      & ~ r1_projred2(A,D,E,F)
                                      & r1_projred2(A,D,F,G)
                                      & ~ r1_incsp_1(A,C,G)
                                      & D != G
                                      & B != C
                                      & r1_incsp_1(A,B,H)
                                      & r1_incsp_1(A,C,H)
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_incsp_1(A))
                                         => ~ ( r1_incsp_1(A,I,H)
                                              & ~ r1_incsp_1(A,I,D)
                                              & ~ r1_incsp_1(A,I,G)
                                              & k5_relat_1(k1_projred1(A,D,F,B),k1_projred1(A,F,E,C)) = k5_relat_1(k1_projred1(A,D,G,I),k1_projred1(A,G,E,C)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t22_projred2,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))
                         => ! [G] :
                              ( m1_subset_1(G,u2_incsp_1(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ~ ( ~ r1_incsp_1(A,B,D)
                                      & ~ r1_incsp_1(A,C,E)
                                      & ~ r1_incsp_1(A,B,F)
                                      & ~ r1_incsp_1(A,C,F)
                                      & ~ r1_projred2(A,D,E,F)
                                      & r1_projred2(A,E,F,G)
                                      & ~ r1_incsp_1(A,B,G)
                                      & E != G
                                      & B != C
                                      & r1_incsp_1(A,B,H)
                                      & r1_incsp_1(A,C,H)
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_incsp_1(A))
                                         => ~ ( r1_incsp_1(A,I,H)
                                              & ~ r1_incsp_1(A,I,E)
                                              & ~ r1_incsp_1(A,I,G)
                                              & k5_relat_1(k1_projred1(A,D,F,B),k1_projred1(A,F,E,C)) = k5_relat_1(k1_projred1(A,D,G,B),k1_projred1(A,G,E,I)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t23_projred2,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,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,u2_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))
                                                     => ( ( r1_incsp_1(A,D,H)
                                                          & r1_incsp_1(A,D,J)
                                                          & r1_incsp_1(A,E,I)
                                                          & r1_incsp_1(A,E,J)
                                                          & r1_incsp_1(A,B,K)
                                                          & r1_incsp_1(A,E,K)
                                                          & r1_incsp_1(A,D,L)
                                                          & r1_incsp_1(A,C,L)
                                                          & r1_incsp_1(A,F,H)
                                                          & r1_incsp_1(A,F,K)
                                                          & r1_incsp_1(A,G,I)
                                                          & r1_incsp_1(A,G,L)
                                                          & r1_incsp_1(A,F,M)
                                                          & r1_incsp_1(A,G,M) )
                                                       => ( r1_incsp_1(A,B,H)
                                                          | r1_incsp_1(A,C,I)
                                                          | r1_incsp_1(A,B,J)
                                                          | r1_incsp_1(A,C,J)
                                                          | r1_incsp_1(A,B,I)
                                                          | r1_incsp_1(A,C,H)
                                                          | r1_projred2(A,H,I,J)
                                                          | k5_relat_1(k1_projred1(A,H,J,B),k1_projred1(A,J,I,C)) = k5_relat_1(k1_projred1(A,H,M,C),k1_projred1(A,M,I,B)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t24_projred2,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))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ~ ( ~ r1_incsp_1(A,B,E)
                                      & ~ r1_incsp_1(A,C,F)
                                      & ~ r1_incsp_1(A,B,G)
                                      & ~ r1_incsp_1(A,C,G)
                                      & B != C
                                      & r1_incsp_1(A,B,H)
                                      & r1_incsp_1(A,C,H)
                                      & r1_incsp_1(A,D,H)
                                      & ~ r1_incsp_1(A,D,E)
                                      & D != C
                                      & ~ r1_projred2(A,E,F,G)
                                      & ! [I] :
                                          ( m1_subset_1(I,u2_incsp_1(A))
                                         => ~ ( r1_projred2(A,E,G,I)
                                              & ~ r1_incsp_1(A,C,I)
                                              & ~ r1_incsp_1(A,D,I)
                                              & k5_relat_1(k1_projred1(A,E,G,B),k1_projred1(A,G,F,C)) = k5_relat_1(k1_projred1(A,E,I,D),k1_projred1(A,I,F,C)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t25_projred2,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))
                             => ! [H] :
                                  ( m1_subset_1(H,u2_incsp_1(A))
                                 => ~ ( ~ r1_incsp_1(A,B,E)
                                      & ~ r1_incsp_1(A,C,F)
                                      & ~ r1_incsp_1(A,B,G)
                                      & ~ r1_incsp_1(A,C,G)
                                      & B != C
                                      & r1_incsp_1(A,B,H)
                                      & r1_incsp_1(A,C,H)
                                      & r1_incsp_1(A,D,H)
                                      & ~ r1_incsp_1(A,D,F)
                                      & D != B
                                      & ~ r1_projred2(A,E,F,G)
                                      & ! [I] :
                                          ( m1_subset_1(I,u2_incsp_1(A))
                                         => ~ ( r1_projred2(A,F,G,I)
                                              & ~ r1_incsp_1(A,B,I)
                                              & ~ r1_incsp_1(A,D,I)
                                              & k5_relat_1(k1_projred1(A,E,G,B),k1_projred1(A,G,F,C)) = k5_relat_1(k1_projred1(A,E,I,B),k1_projred1(A,I,F,D)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_projred2,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_projred2(B,A)
         => ( v1_funct_1(B)
            & m2_relset_1(B,u1_incsp_1(A),u1_incsp_1(A)) ) ) ) )).

fof(existence_m1_projred2,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_projred2(B,A) ) )).

fof(dt_k1_projred2,axiom,(
    ! [A,B] :
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A)
        & m1_subset_1(B,u2_incsp_1(A)) )
     => m1_subset_1(k1_projred2(A,B),k1_zfmisc_1(u1_incsp_1(A))) ) )).

fof(d2_projred2,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))
         => k1_projred2(A,B) = a_2_0_projred2(A,B) ) ) )).

fof(fraenkel_a_2_0_projred2,axiom,(
    ! [A,B,C] :
      ( ( v1_incproj(B)
        & v2_incproj(B)
        & v3_incproj(B)
        & v4_incproj(B)
        & v5_incproj(B)
        & l1_incsp_1(B)
        & m1_subset_1(C,u2_incsp_1(B)) )
     => ( r2_hidden(A,a_2_0_projred2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_incsp_1(B))
            & A = D
            & r1_incsp_1(B,D,C) ) ) ) )).
%------------------------------------------------------------------------------