% Mizar problem: t5_projred2,projred2,101,63 
fof(t5_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,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) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,cc1_relset_1,cc2_funct_1,d1_projred1,d2_projred2,dt_k1_funct_1,dt_k1_projred1,dt_k1_projred2,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,fraenkel_a_2_0_projred2,fraenkel_a_2_1_projred2,rc1_funct_1,rc2_funct_1,rc3_funct_1,redefinition_k4_relset_1,redefinition_m2_relset_1,reflexivity_r1_tarski,t1_subset,t2_subset,t2_tarski,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(projred2,t5_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(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(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) ) ) ),
    file(projred2,d2_projred2),
    []).

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_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))) ) ),
    file(projred2,k1_projred2),
    []).

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(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) ) ) ) ),
    file(projred2,a_2_0_projred2),
    []).

fof(fraenkel_a_2_1_projred2,axiom,(
    ! [A,B,C] : 
      ( ( v1_incproj(B)
        & v2_incproj(B)
        & v3_incproj(B)
        & v4_incproj(B)
        & v5_incproj(B)
        & v6_incproj(B)
        & v10_incproj(B)
        & l1_incsp_1(B)
        & m1_subset_1(C,u2_incsp_1(B)) )
     => ( r2_hidden(A,a_2_1_projred2(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,u1_incsp_1(B))
            & A = D
            & r1_incsp_1(B,D,C) ) ) ) ),
    file(projred2,a_2_1_projred2),
    []).

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

fof(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

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),
    []).