% Mizar problem: t17_projred2,projred2,374,46 
fof(t17_projred2,conjecture,(
    ! [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) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,d1_projred2,d7_incproj,d8_incproj,d9_incproj,dt_k1_xboole_0,dt_l1_incsp_1,dt_m1_subset_1,dt_u1_incsp_1,dt_u2_incsp_1,existence_l1_incsp_1,existence_m1_subset_1,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(projred2,t17_projred2)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_funct_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    []).

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) ) ) ) ) ) ) ),
    file(projred2,d1_projred2),
    []).

fof(d7_incproj,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))
                 => ( r3_incproj(A,B,C,D)
                  <=> ( r1_incsp_1(A,B,D)
                      & r1_incsp_1(A,C,D) ) ) ) ) ) ) ),
    file(incproj,d7_incproj),
    []).

fof(d8_incproj,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))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_incsp_1(A))
                     => ( r4_incproj(A,B,C,D,E)
                      <=> ( r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,C,D)
                          & r1_incsp_1(A,E,D) ) ) ) ) ) ) ) ),
    file(incproj,d8_incproj),
    []).

fof(d9_incproj,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v1_incproj(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 ) ) ) ) ) ) ) ),
    file(incproj,d9_incproj),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_l1_incsp_1,axiom,(
    $true ),
    file(incsp_1,l1_incsp_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_u1_incsp_1,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ~ v1_xboole_0(u1_incsp_1(A)) ) ),
    file(incsp_1,u1_incsp_1),
    []).

fof(dt_u2_incsp_1,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ~ v1_xboole_0(u2_incsp_1(A)) ) ),
    file(incsp_1,u2_incsp_1),
    []).

fof(existence_l1_incsp_1,axiom,(
    ? [A] : l1_incsp_1(A) ),
    file(incsp_1,l1_incsp_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).