% Mizar problem: t8_heyting2,heyting2,180,76 
fof(t8_heyting2,conjecture,(
    ! [A,B,C] : 
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => ! [D] : 
          ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
         => ( k3_substlat(A,B,k4_substlat(A,B,C,D)) = C
           => ! [E] : ~ ( r2_hidden(E,C)
                & ! [F] : ~ ( r2_hidden(F,D)
                    & r1_tarski(F,E) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finseq_1,cc1_finsub_1,cc1_fraenkel,cc1_funct_1,cc1_substlat,cc2_finsub_1,cc2_funct_1,cc2_substlat,cc3_finsub_1,commutativity_k2_xboole_0,dt_k1_substlat,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k3_substlat,dt_k4_partfun1,dt_k4_substlat,dt_k5_finsub_1,dt_m1_subset_1,dt_m2_subset_1,existence_m1_subset_1,existence_m2_subset_1,fc1_finsub_1,fc1_subset_1,fc1_substlat,fc1_wellfnd1,fc2_finseq_1,fc2_finsub_1,idempotence_k2_xboole_0,rc1_finseq_1,rc1_fraenkel,rc1_funct_1,rc1_heyting2,rc1_setwiseo,rc1_subset_1,rc1_substlat,rc2_funct_1,rc2_setwiseo,rc2_subset_1,rc2_substlat,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,redefinition_m2_subset_1,redefinition_r1_heyting1,reflexivity_r1_heyting1,reflexivity_r1_tarski,t15_substlat,t1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t7_xboole_1,t8_boole,t8_substlat]),
    [file(heyting2,t8_heyting2)]).

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

fof(cc1_finseq_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) ) ) ),
    file(finseq_1,cc1_finseq_1),
    []).

fof(cc1_finsub_1,axiom,(
    ! [A] : 
      ( v4_finsub_1(A)
     => ( v1_finsub_1(A)
        & v3_finsub_1(A) ) ) ),
    file(finsub_1,cc1_finsub_1),
    []).

fof(cc1_fraenkel,axiom,(
    ! [A] : 
      ( v1_fraenkel(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) ),
    file(fraenkel,cc1_fraenkel),
    []).

fof(cc1_funct_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    []).

fof(cc1_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
     => v1_finset_1(C) ) ),
    file(substlat,cc1_substlat),
    []).

fof(cc2_finsub_1,axiom,(
    ! [A] : 
      ( ( v1_finsub_1(A)
        & v3_finsub_1(A) )
     => v4_finsub_1(A) ) ),
    file(finsub_1,cc2_finsub_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(cc2_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
     => ( v1_finset_1(C)
        & v1_fraenkel(C) ) ) ),
    file(substlat,cc2_substlat),
    []).

fof(cc3_finsub_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k5_finsub_1(A))
     => v1_finset_1(B) ) ),
    file(finsub_1,cc3_finsub_1),
    []).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k1_substlat,axiom,(
    ! [A,B] : m1_subset_1(k1_substlat(A,B),k1_zfmisc_1(k5_finsub_1(k4_partfun1(A,B)))) ),
    file(substlat,k1_substlat),
    []).

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_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k3_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => m2_subset_1(k3_substlat(A,B,C),k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B)) ) ),
    file(substlat,k3_substlat),
    []).

fof(dt_k4_partfun1,axiom,(
    $true ),
    file(partfun1,k4_partfun1),
    []).

fof(dt_k4_substlat,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
     => m1_subset_1(k4_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) ),
    file(substlat,k4_substlat),
    []).

fof(dt_k5_finsub_1,axiom,(
    ! [A] : v4_finsub_1(k5_finsub_1(A)) ),
    file(finsub_1,k5_finsub_1),
    []).

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

fof(dt_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    []).

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

fof(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    []).

fof(fc1_finsub_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v1_finsub_1(k1_zfmisc_1(A))
      & v3_finsub_1(k1_zfmisc_1(A))
      & v4_finsub_1(k1_zfmisc_1(A)) ) ),
    file(finsub_1,fc1_finsub_1),
    []).

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

fof(fc1_substlat,axiom,(
    ! [A,B] : ~ v1_xboole_0(k1_substlat(A,B)) ),
    file(substlat,fc1_substlat),
    []).

fof(fc1_wellfnd1,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(k4_partfun1(A,B))
      & v1_fraenkel(k4_partfun1(A,B)) ) ),
    file(wellfnd1,fc1_wellfnd1),
    []).

fof(fc2_finseq_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_finset_1(k1_xboole_0)
    & v1_finseq_1(k1_xboole_0) ),
    file(finseq_1,fc2_finseq_1),
    []).

fof(fc2_finsub_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k5_finsub_1(A))
      & v1_finsub_1(k5_finsub_1(A))
      & v3_finsub_1(k5_finsub_1(A))
      & v4_finsub_1(k5_finsub_1(A)) ) ),
    file(finsub_1,fc2_finsub_1),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(rc1_finseq_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc1_finseq_1),
    []).

fof(rc1_fraenkel,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_fraenkel(A) ) ),
    file(fraenkel,rc1_fraenkel),
    []).

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

fof(rc1_heyting2,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & v1_fraenkel(A) ) ),
    file(heyting2,rc1_heyting2),
    []).

fof(rc1_setwiseo,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k5_finsub_1(A))
      & v1_xboole_0(B)
      & v1_finset_1(B) ) ),
    file(setwiseo,rc1_setwiseo),
    []).

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(rc1_substlat,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
      & ~ v1_xboole_0(C) ) ),
    file(substlat,rc1_substlat),
    []).

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(rc2_setwiseo,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k5_finsub_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(setwiseo,rc2_setwiseo),
    []).

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(rc2_substlat,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_subset_1(C,k4_partfun1(A,B))
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_finset_1(C) ) ),
    file(substlat,rc2_substlat),
    []).

fof(rc3_finseq_1,axiom,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc3_finseq_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(rc4_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    []).

fof(rc6_finseq_1,axiom,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc6_finseq_1),
    []).

fof(rc7_finseq_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc7_finseq_1),
    []).

fof(rc8_finseq_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc8_finseq_1),
    []).

fof(redefinition_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(subset_1,m2_subset_1),
    []).

fof(redefinition_r1_heyting1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_finsub_1(A))
        & m1_subset_1(C,k5_finsub_1(A)) )
     => ( r1_heyting1(A,B,C)
      <=> r1_tarski(B,C) ) ) ),
    file(heyting1,r1_heyting1),
    []).

fof(reflexivity_r1_heyting1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_finsub_1(A))
        & m1_subset_1(C,k5_finsub_1(A)) )
     => r1_heyting1(A,B,B) ) ),
    file(heyting1,r1_heyting1),
    []).

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

fof(t15_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] : 
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] : ~ ( r2_hidden(E,k4_substlat(A,B,C,D))
              & ! [F,G] : ~ ( r2_hidden(F,C)
                  & r2_hidden(G,D)
                  & E = k2_xboole_0(F,G) ) ) ) ) ),
    file(substlat,t15_substlat),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

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(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(t7_xboole_1,axiom,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    []).

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

fof(t8_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => r1_tarski(k3_substlat(A,B,C),C) ) ),
    file(substlat,t8_substlat),
    []).