% Mizar problem: t15_incproj,incproj,261,21 
fof(t15_incproj,conjecture,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ~ ! [B] : 
            ( m1_subset_1(B,u1_incsp_1(k3_incproj(A)))
           => ! [C] : 
                ( m1_subset_1(C,u2_incsp_1(k3_incproj(A)))
               => r1_incsp_1(k3_incproj(A),B,C) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[abstractness_v1_incsp_1,antisymmetry_r2_hidden,cc1_relset_1,d1_incproj,d3_incproj,d6_collsp,d7_collsp,dt_g1_incsp_1,dt_k1_collsp,dt_k1_incproj,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_incproj,dt_k2_zfmisc_1,dt_k3_incproj,dt_l1_collsp,dt_l1_incsp_1,dt_l1_struct_0,dt_m1_incproj,dt_m1_relset_1,dt_m1_subset_1,dt_m2_collsp,dt_m2_relset_1,dt_u1_incsp_1,dt_u1_struct_0,dt_u2_incsp_1,dt_u3_incsp_1,existence_l1_collsp,existence_l1_incsp_1,existence_l1_struct_0,existence_m1_incproj,existence_m1_relset_1,existence_m1_subset_1,existence_m2_collsp,existence_m2_relset_1,fc1_incproj,fc1_struct_0,fc1_subset_1,fc2_incproj,fc4_subset_1,fraenkel_a_1_0_incproj,free_g1_incsp_1,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,redefinition_m1_incproj,redefinition_m2_relset_1,reflexivity_r1_tarski,t17_collsp,t1_subset,t2_subset,t2_tarski,t3_subset,t4_subset,t5_subset,t6_boole,t6_incproj,t7_boole,t7_collsp,t8_boole,t9_incproj]),
    [file(incproj,t15_incproj)]).

fof(abstractness_v1_incsp_1,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ( v1_incsp_1(A)
       => A = g1_incsp_1(u1_incsp_1(A),u2_incsp_1(A),u3_incsp_1(A)) ) ) ),
    file(incsp_1,v1_incsp_1),
    []).

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

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(d1_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => k1_incproj(A) = a_1_0_incproj(A) ) ),
    file(incproj,d1_incproj),
    []).

fof(d3_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => k3_incproj(A) = g1_incsp_1(u1_struct_0(A),k1_incproj(A),k2_incproj(A)) ) ),
    file(incproj,d3_incproj),
    []).

fof(d6_collsp,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_collsp(A) )
     => ( v4_collsp(A)
      <=> ~ ! [B] : 
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(A))
                     => r1_collsp(A,B,C,D) ) ) ) ) ) ),
    file(collsp,d6_collsp),
    []).

fof(d7_collsp,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m2_collsp(B,A)
        <=> ? [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
              & ? [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                  & C != D
                  & B = k1_collsp(A,C,D) ) ) ) ) ),
    file(collsp,d7_collsp),
    []).

fof(dt_g1_incsp_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_relset_1(C,A,B) )
     => ( v1_incsp_1(g1_incsp_1(A,B,C))
        & l1_incsp_1(g1_incsp_1(A,B,C)) ) ) ),
    file(incsp_1,g1_incsp_1),
    []).

fof(dt_k1_collsp,axiom,(
    $true ),
    file(collsp,k1_collsp),
    []).

fof(dt_k1_incproj,axiom,(
    $true ),
    file(incproj,k1_incproj),
    []).

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_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => m2_relset_1(k2_incproj(A),u1_struct_0(A),k1_incproj(A)) ) ),
    file(incproj,k2_incproj),
    []).

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

fof(dt_k3_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => l1_incsp_1(k3_incproj(A)) ) ),
    file(incproj,k3_incproj),
    []).

fof(dt_l1_collsp,axiom,(
    ! [A] : 
      ( l1_collsp(A)
     => l1_struct_0(A) ) ),
    file(collsp,l1_collsp),
    []).

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

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    []).

fof(dt_m1_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_incproj(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ),
    file(incproj,m1_incproj),
    []).

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_collsp,axiom,(
    $true ),
    file(collsp,m2_collsp),
    []).

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_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

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(dt_u3_incsp_1,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => m2_relset_1(u3_incsp_1(A),u1_incsp_1(A),u2_incsp_1(A)) ) ),
    file(incsp_1,u3_incsp_1),
    []).

fof(existence_l1_collsp,axiom,(
    ? [A] : l1_collsp(A) ),
    file(collsp,l1_collsp),
    []).

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

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_m1_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ? [B] : m1_incproj(B,A) ) ),
    file(incproj,m1_incproj),
    []).

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_collsp,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ? [B] : m2_collsp(B,A) ) ),
    file(collsp,m2_collsp),
    []).

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

fof(fc1_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ~ v1_xboole_0(k1_incproj(A)) ) ),
    file(incproj,fc1_incproj),
    []).

fof(fc1_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc2_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => v1_incsp_1(k3_incproj(A)) ) ),
    file(incproj,fc2_incproj),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

fof(fraenkel_a_1_0_incproj,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(B)
        & v2_collsp(B)
        & v3_collsp(B)
        & v4_collsp(B)
        & l1_collsp(B) )
     => ( r2_hidden(A,a_1_0_incproj(B))
      <=> ? [C] : 
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
            & A = C
            & m1_incproj(C,B) ) ) ) ),
    file(incproj,a_1_0_incproj),
    []).

fof(free_g1_incsp_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_relset_1(C,A,B) )
     => ! [D,E,F] : 
          ( g1_incsp_1(A,B,C) = g1_incsp_1(D,E,F)
         => ( A = D
            & B = E
            & C = F ) ) ) ),
    file(incsp_1,g1_incsp_1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc3_struct_0,axiom,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    []).

fof(rc5_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    []).

fof(redefinition_m1_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_incproj(B,A)
        <=> m2_collsp(B,A) ) ) ),
    file(incproj,m1_incproj),
    []).

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(t17_collsp,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_collsp(A,B,C,D)
                  <=> r2_hidden(D,k1_collsp(A,B,C)) ) ) ) ) ) ),
    file(collsp,t17_collsp),
    []).

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(t6_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_incproj(B,A)
        <=> m1_subset_1(B,u2_incsp_1(k3_incproj(A))) ) ) ),
    file(incproj,t6_incproj),
    []).

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

fof(t7_collsp,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ~ ( B != C
                        & B != D
                        & C != D )
                   => r1_collsp(A,B,C,D) ) ) ) ) ) ),
    file(collsp,t7_collsp),
    []).

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

fof(t9_incproj,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_incsp_1(k3_incproj(A)))
         => ! [C] : 
              ( m1_subset_1(C,u2_incsp_1(k3_incproj(A)))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_incproj(E,A)
                     => ( ( B = D
                          & C = E )
                       => ( r1_incsp_1(k3_incproj(A),B,C)
                        <=> r2_hidden(D,E) ) ) ) ) ) ) ) ),
    file(incproj,t9_incproj),
    []).