% Mizar problem: t31_sheffer1,sheffer1,1036,13 
fof(t31_sheffer1,conjecture,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v10_sheffer1(A)
        & l3_sheffer1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => B = k6_sheffer1(A,k6_sheffer1(A,B)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[rc2_tex_2,cc1_tex_2,rc3_tex_2,rc4_tex_2,existence_l1_lattices,existence_l1_robbins1,existence_l2_lattices,dt_k1_zfmisc_1,dt_l1_lattices,dt_l1_robbins1,dt_l2_lattices,rc5_struct_0,cc2_tex_2,existence_l2_robbins1,existence_l3_lattices,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_binop_1,dt_k2_zfmisc_1,dt_l2_robbins1,dt_l3_lattices,dt_m1_relset_1,dt_m2_relset_1,existence_l1_struct_0,existence_l3_robbins1,redefinition_k2_binop_1,dt_k2_binop_1,dt_l1_struct_0,dt_l3_robbins1,dt_u1_sheffer1,rc3_struct_0,fc1_struct_0,existence_l1_sheffer1,existence_l3_sheffer1,existence_m1_subset_1,dt_k5_sheffer1,dt_k6_sheffer1,dt_l1_sheffer1,dt_l3_sheffer1,dt_m1_subset_1,dt_u1_struct_0,rc5_sheffer1,rc7_sheffer1,d11_sheffer1,d16_sheffer1,d13_sheffer1]),
    [file(sheffer1,t31_sheffer1)]).

fof(rc2_tex_2,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ),
    file(tex_2,rc2_tex_2),
    []).

fof(cc1_tex_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( ~ v1_tex_2(B,k1_zfmisc_1(A))
           => ~ v1_xboole_0(B) ) ) ) ),
    file(tex_2,cc1_tex_2),
    []).

fof(rc3_tex_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ),
    file(tex_2,rc3_tex_2),
    []).

fof(rc4_tex_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ),
    file(tex_2,rc4_tex_2),
    []).

fof(existence_l1_lattices,axiom,(
    ? [A] : l1_lattices(A) ),
    file(lattices,l1_lattices),
    []).

fof(existence_l1_robbins1,axiom,(
    ? [A] : l1_robbins1(A) ),
    file(robbins1,l1_robbins1),
    []).

fof(existence_l2_lattices,axiom,(
    ? [A] : l2_lattices(A) ),
    file(lattices,l2_lattices),
    []).

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

fof(dt_l1_lattices,axiom,(
    ! [A] : 
      ( l1_lattices(A)
     => l1_struct_0(A) ) ),
    file(lattices,l1_lattices),
    []).

fof(dt_l1_robbins1,axiom,(
    ! [A] : 
      ( l1_robbins1(A)
     => l1_struct_0(A) ) ),
    file(robbins1,l1_robbins1),
    []).

fof(dt_l2_lattices,axiom,(
    ! [A] : 
      ( l2_lattices(A)
     => l1_struct_0(A) ) ),
    file(lattices,l2_lattices),
    []).

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(cc2_tex_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_xboole_0(B)
           => v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ),
    file(tex_2,cc2_tex_2),
    []).

fof(existence_l2_robbins1,axiom,(
    ? [A] : l2_robbins1(A) ),
    file(robbins1,l2_robbins1),
    []).

fof(existence_l3_lattices,axiom,(
    ? [A] : l3_lattices(A) ),
    file(lattices,l3_lattices),
    []).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_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(dt_k1_binop_1,axiom,(
    $true ),
    file(binop_1,k1_binop_1),
    []).

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

fof(dt_l2_robbins1,axiom,(
    ! [A] : 
      ( l2_robbins1(A)
     => ( l2_lattices(A)
        & l1_robbins1(A) ) ) ),
    file(robbins1,l2_robbins1),
    []).

fof(dt_l3_lattices,axiom,(
    ! [A] : 
      ( l3_lattices(A)
     => ( l1_lattices(A)
        & l2_lattices(A) ) ) ),
    file(lattices,l3_lattices),
    []).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_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(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_l3_robbins1,axiom,(
    ? [A] : l3_robbins1(A) ),
    file(robbins1,l3_robbins1),
    []).

fof(redefinition_k2_binop_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B) )
     => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ),
    file(binop_1,k2_binop_1),
    []).

fof(dt_k2_binop_1,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B) )
     => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ),
    file(binop_1,k2_binop_1),
    []).

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

fof(dt_l3_robbins1,axiom,(
    ! [A] : 
      ( l3_robbins1(A)
     => ( l2_robbins1(A)
        & l3_lattices(A) ) ) ),
    file(robbins1,l3_robbins1),
    []).

fof(dt_u1_sheffer1,axiom,(
    ! [A] : 
      ( l1_sheffer1(A)
     => ( v1_funct_1(u1_sheffer1(A))
        & v1_funct_2(u1_sheffer1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
        & m2_relset_1(u1_sheffer1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ),
    file(sheffer1,u1_sheffer1),
    []).

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

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(existence_l1_sheffer1,axiom,(
    ? [A] : l1_sheffer1(A) ),
    file(sheffer1,l1_sheffer1),
    []).

fof(existence_l3_sheffer1,axiom,(
    ? [A] : l3_sheffer1(A) ),
    file(sheffer1,l3_sheffer1),
    []).

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

fof(dt_k5_sheffer1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_sheffer1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k5_sheffer1(A,B,C),u1_struct_0(A)) ) ),
    file(sheffer1,k5_sheffer1),
    []).

fof(dt_k6_sheffer1,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l1_sheffer1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k6_sheffer1(A,B),u1_struct_0(A)) ) ),
    file(sheffer1,k6_sheffer1),
    []).

fof(dt_l1_sheffer1,axiom,(
    ! [A] : 
      ( l1_sheffer1(A)
     => l1_struct_0(A) ) ),
    file(sheffer1,l1_sheffer1),
    []).

fof(dt_l3_sheffer1,axiom,(
    ! [A] : 
      ( l3_sheffer1(A)
     => ( l1_sheffer1(A)
        & l3_robbins1(A) ) ) ),
    file(sheffer1,l3_sheffer1),
    []).

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

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

fof(rc5_sheffer1,axiom,(
    ? [A] : 
      ( l1_sheffer1(A)
      & ~ v3_struct_0(A) ) ),
    file(sheffer1,rc5_sheffer1),
    []).

fof(rc7_sheffer1,axiom,(
    ? [A] : 
      ( l3_sheffer1(A)
      & ~ v3_struct_0(A) ) ),
    file(sheffer1,rc7_sheffer1),
    []).

fof(d11_sheffer1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_sheffer1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_sheffer1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_sheffer1(A),B,C) ) ) ) ),
    file(sheffer1,d11_sheffer1),
    []).

fof(d16_sheffer1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_sheffer1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_sheffer1(A,B) = k5_sheffer1(A,B,B) ) ) ),
    file(sheffer1,d16_sheffer1),
    []).

fof(d13_sheffer1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_sheffer1(A) )
     => ( v10_sheffer1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k5_sheffer1(A,k5_sheffer1(A,B,B),k5_sheffer1(A,B,B)) = B ) ) ) ),
    file(sheffer1,d13_sheffer1),
    []).