% Mizar problem: t4_projpl_1,projpl_1,125,29 
fof(t4_projpl_1,conjecture,(
    ! [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) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[d10_incproj,d11_incproj,d12_incproj,d13_incproj,d2_projpl_1,d4_projpl_1,d5_incproj,d7_incproj,d8_incproj,d9_incproj,dt_l1_incsp_1,dt_m1_subset_1,dt_u1_incsp_1,dt_u2_incsp_1,existence_l1_incsp_1,existence_m1_subset_1,rc1_xboole_0,rc2_xboole_0]),
    [file(projpl_1,t4_projpl_1)]).

fof(d10_incproj,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v2_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))
                    & r1_incsp_1(A,B,D)
                    & r1_incsp_1(A,C,D) ) ) ) ) ) ),
    file(incproj,d10_incproj),
    []).

fof(d11_incproj,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v3_incproj(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) ) ) ) ) ),
    file(incproj,d11_incproj),
    []).

fof(d12_incproj,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v4_incproj(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))
                        & C != D
                        & D != E
                        & E != C
                        & r1_incsp_1(A,C,B)
                        & r1_incsp_1(A,D,B)
                        & r1_incsp_1(A,E,B) ) ) ) ) ) ) ),
    file(incproj,d12_incproj),
    []).

fof(d13_incproj,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v5_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,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))
                                               => ~ ( r1_incsp_1(A,B,H)
                                                    & r1_incsp_1(A,C,H)
                                                    & r1_incsp_1(A,D,I)
                                                    & r1_incsp_1(A,E,I)
                                                    & r1_incsp_1(A,F,H)
                                                    & r1_incsp_1(A,F,I)
                                                    & r1_incsp_1(A,B,J)
                                                    & r1_incsp_1(A,D,J)
                                                    & r1_incsp_1(A,C,K)
                                                    & r1_incsp_1(A,E,K)
                                                    & ~ r1_incsp_1(A,F,J)
                                                    & ~ r1_incsp_1(A,F,K)
                                                    & H != I
                                                    & ! [L] : 
                                                        ( m1_subset_1(L,u1_incsp_1(A))
                                                       => ~ ( r1_incsp_1(A,L,J)
                                                            & r1_incsp_1(A,L,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(incproj,d13_incproj),
    []).

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) ) ) ) ) ) ) ),
    file(projpl_1,d2_projpl_1),
    []).

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 ) ) ) ) ) ) ) ),
    file(projpl_1,d4_projpl_1),
    []).

fof(d5_incproj,axiom,(
    ! [A,B,C] : 
      ( r1_incproj(A,B,C)
    <=> ( A != B
        & B != C
        & C != A ) ) ),
    file(incproj,d5_incproj),
    []).

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_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(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).