% Mizar problem: t26_projred1,projred1,659,11 
fof(t26_projred1,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,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 ) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,cc1_relset_1,cc2_funct_1,d1_projred1,d9_incproj,dt_k1_funct_1,dt_k1_projred1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_relset_1,dt_l1_incsp_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_incsp_1,dt_u2_incsp_1,existence_l1_incsp_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,rc1_funct_1,rc2_funct_1,rc3_funct_1,redefinition_k4_relset_1,redefinition_m2_relset_1,reflexivity_r1_tarski,t1_subset,t23_projred1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(projred1,t26_projred1)]).

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(cc1_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    []).

fof(cc2_funct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    []).

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) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(projred1,d1_projred1),
    []).

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_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

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)) ) ) ),
    file(projred1,k1_projred1),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

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

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(relset_1,k4_relset_1),
    []).

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

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

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

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_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_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

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

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(rc1_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    []).

fof(rc2_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    []).

fof(rc3_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    []).

fof(redefinition_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) ),
    file(relset_1,k4_relset_1),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

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

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) ) ) ) ) ) ) ) ) ),
    file(projred1,t23_projred1),
    []).

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

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_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),
    []).