% Mizar problem: t30_projpl_1,projpl_1,812,21 
fof(t30_projpl_1,conjecture,(
    ! [A] : 
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_incsp_1(A))
                 => ! [E] : 
                      ( m1_subset_1(E,u2_incsp_1(A))
                     => ( ( r1_incsp_1(A,C,E)
                          & r1_incsp_1(A,D,E) )
                       => ( C = D
                          | r1_incsp_1(A,B,E)
                          | ( k1_projpl_1(A,C,B) != k1_projpl_1(A,D,B)
                            & k1_projpl_1(A,B,C) != k1_projpl_1(A,B,D) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[d2_projpl_1,d9_projpl_1,dt_k1_projpl_1,dt_k2_projpl_1,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,t16_projpl_1]),
    [file(projpl_1,t30_projpl_1)]).

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(d9_projpl_1,axiom,(
    ! [A] : 
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u2_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u2_incsp_1(A))
             => ( B != C
               => ! [D] : 
                    ( m1_subset_1(D,u1_incsp_1(A))
                   => ( D = k2_projpl_1(A,B,C)
                    <=> r2_projpl_1(A,D,B,C) ) ) ) ) ) ) ),
    file(projpl_1,d9_projpl_1),
    []).

fof(dt_k1_projpl_1,axiom,(
    ! [A,B,C] : 
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A)
        & m1_subset_1(B,u1_incsp_1(A))
        & m1_subset_1(C,u1_incsp_1(A)) )
     => m1_subset_1(k1_projpl_1(A,B,C),u2_incsp_1(A)) ) ),
    file(projpl_1,k1_projpl_1),
    []).

fof(dt_k2_projpl_1,axiom,(
    ! [A,B,C] : 
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & v6_incproj(A)
        & l1_incsp_1(A)
        & m1_subset_1(B,u2_incsp_1(A))
        & m1_subset_1(C,u2_incsp_1(A)) )
     => m1_subset_1(k2_projpl_1(A,B,C),u1_incsp_1(A)) ) ),
    file(projpl_1,k2_projpl_1),
    []).

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),
    []).

fof(t16_projpl_1,axiom,(
    ! [A] : 
      ( ( v1_incproj(A)
        & v2_incproj(A)
        & v3_incproj(A)
        & v4_incproj(A)
        & v5_incproj(A)
        & l1_incsp_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] : 
                  ( m1_subset_1(D,u2_incsp_1(A))
                 => ( B != C
                   => ( r1_incsp_1(A,B,k1_projpl_1(A,B,C))
                      & r1_incsp_1(A,C,k1_projpl_1(A,B,C))
                      & k1_projpl_1(A,B,C) = k1_projpl_1(A,C,B)
                      & ( ( r1_incsp_1(A,B,D)
                          & r1_incsp_1(A,C,D) )
                       => D = k1_projpl_1(A,B,C) ) ) ) ) ) ) ) ),
    file(projpl_1,t16_projpl_1),
    []).