% Mizar problem: t11_knaster,knaster,309,19 
fof(t11_knaster,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,B,A)
                    & m2_relset_1(D,B,A) )
                 => ? [E,F,G,H] : 
                      ( r1_xboole_0(E,F)
                      & r1_xboole_0(G,H)
                      & k2_xboole_0(E,F) = A
                      & k2_xboole_0(G,H) = B
                      & k2_funct_2(A,B,C,E) = G
                      & k2_funct_2(B,A,D,H) = F ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,cc1_ordinal1,cc1_relset_1,cc2_funct_1,cc2_ordinal1,cc3_ordinal1,commutativity_k2_xboole_0,commutativity_k3_xboole_0,d12_cohsp_1,d3_abian,d4_knaster,dt_k1_funct_1,dt_k1_knaster,dt_k1_relat_1,dt_k1_setfam_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_funct_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_xboole_0,dt_k4_relset_1,dt_k4_xboole_0,dt_k8_funct_2,dt_k9_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_subset_1,fc2_ordinal1,fc4_subset_1,fraenkel_a_2_0_knaster,idempotence_k2_xboole_0,idempotence_k3_xboole_0,rc1_funct_1,rc1_knaster,rc1_ordinal1,rc1_subset_1,rc2_funct_1,rc2_ordinal1,rc2_subset_1,rc3_funct_1,rc3_ordinal1,rc4_funct_1,redefinition_k2_funct_2,redefinition_k4_relset_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_r2_abian,reflexivity_r1_tarski,s2_funct_2__e2_14__knaster,symmetry_r1_xboole_0,t13_prob_1,t156_relat_1,t1_boole,t1_subset,t28_xboole_1,t2_boole,t2_subset,t2_tarski,t34_xboole_1,t36_xboole_1,t3_boole,t3_subset,t45_xboole_1,t48_xboole_1,t4_boole,t4_subset,t5_subset,t6_boole,t6_knaster,t7_boole,t8_boole]),
    [file(knaster,t11_knaster)]).

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_ordinal1,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    []).

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(cc2_ordinal1,axiom,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    []).

fof(cc3_ordinal1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    []).

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(commutativity_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    []).

fof(d12_cohsp_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v6_cohsp_1(A)
      <=> ! [B,C] : 
            ( ( r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A))
              & r1_tarski(B,C) )
           => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ),
    file(cohsp_1,d12_cohsp_1),
    []).

fof(d3_abian,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_abian(A,B)
      <=> ( r2_hidden(A,k1_relat_1(B))
          & A = k1_funct_1(B,A) ) ) ) ),
    file(abian,d3_abian),
    []).

fof(d4_knaster,axiom,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
        & v6_cohsp_1(B)
        & m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
     => k1_knaster(A,B) = k1_setfam_1(a_2_0_knaster(A,B)) ) ),
    file(knaster,d4_knaster),
    []).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

fof(dt_k1_knaster,axiom,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
        & v6_cohsp_1(B)
        & m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
     => m1_subset_1(k1_knaster(A,B),k1_zfmisc_1(A)) ) ),
    file(knaster,k1_knaster),
    []).

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

fof(dt_k1_setfam_1,axiom,(
    $true ),
    file(setfam_1,k1_setfam_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_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B) )
     => m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ),
    file(funct_2,k2_funct_2),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

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

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    []).

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_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    []).

fof(dt_k8_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ),
    file(funct_2,k8_funct_2),
    []).

fof(dt_k9_relat_1,axiom,(
    $true ),
    file(relat_1,k9_relat_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(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(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc2_ordinal1,axiom,
    ( 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_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    []).

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_2_0_knaster,axiom,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
        & v6_cohsp_1(C)
        & m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_2_0_knaster(B,C))
      <=> ? [D] : 
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = D
            & r1_tarski(k8_funct_2(k1_zfmisc_1(B),k1_zfmisc_1(B),C,D),D) ) ) ) ),
    file(knaster,a_2_0_knaster),
    []).

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

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

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

fof(rc1_knaster,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ? [C] : 
          ( m1_relset_1(C,A,B)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,A,B)
          & v6_cohsp_1(C) ) ) ),
    file(knaster,rc1_knaster),
    []).

fof(rc1_ordinal1,axiom,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    []).

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_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    []).

fof(rc2_ordinal1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    []).

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_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    []).

fof(rc3_ordinal1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    []).

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(redefinition_k2_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B) )
     => k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ),
    file(funct_2,k2_funct_2),
    []).

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_k8_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ),
    file(funct_2,k8_funct_2),
    []).

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(redefinition_r2_abian,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( r2_abian(A,B,C)
      <=> r1_abian(B,C) ) ) ),
    file(abian,r2_abian),
    []).

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

fof(s2_funct_2__e2_14__knaster,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B)
        & v1_funct_1(D)
        & v1_funct_2(D,B,A)
        & m2_relset_1(D,B,A) )
     => ( ! [E] : 
            ( r2_hidden(E,k1_zfmisc_1(A))
           => r2_hidden(k4_xboole_0(A,k2_funct_2(B,A,D,k4_xboole_0(B,k2_funct_2(A,B,C,E)))),k1_zfmisc_1(A)) )
       => ? [E] : 
            ( v1_funct_1(E)
            & v1_funct_2(E,k1_zfmisc_1(A),k1_zfmisc_1(A))
            & m2_relset_1(E,k1_zfmisc_1(A),k1_zfmisc_1(A))
            & ! [F] : 
                ( r2_hidden(F,k1_zfmisc_1(A))
               => k1_funct_1(E,F) = k4_xboole_0(A,k2_funct_2(B,A,D,k4_xboole_0(B,k2_funct_2(A,B,C,F)))) ) ) ) ) ),
    file(knaster,s2_funct_2__e2_14__knaster),
    []).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    []).

fof(t13_prob_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & v2_finsub_1(B)
        & v1_prob_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C,D] : 
          ( r1_xboole_0(k4_xboole_0(C,D),D)
          & ( ( r2_hidden(C,B)
              & r2_hidden(D,B) )
           => r2_hidden(k2_xboole_0(k4_xboole_0(C,D),D),B) ) ) ) ),
    file(prob_1,t13_prob_1),
    []).

fof(t156_relat_1,axiom,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( r1_tarski(A,B)
       => r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ),
    file(relat_1,t156_relat_1),
    []).

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(t28_xboole_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    file(xboole_1,t28_xboole_1),
    []).

fof(t2_boole,axiom,(
    ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ),
    file(boole,t2_boole),
    []).

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(t34_xboole_1,axiom,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(C,B),k4_xboole_0(C,A)) ) ),
    file(xboole_1,t34_xboole_1),
    []).

fof(t36_xboole_1,axiom,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    []).

fof(t3_boole,axiom,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    []).

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

fof(t45_xboole_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => B = k2_xboole_0(A,k4_xboole_0(B,A)) ) ),
    file(xboole_1,t45_xboole_1),
    []).

fof(t48_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(A,k4_xboole_0(A,B)) = k3_xboole_0(A,B) ),
    file(xboole_1,t48_xboole_1),
    []).

fof(t4_boole,axiom,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    []).

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_knaster,axiom,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
        & v6_cohsp_1(B)
        & m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
     => r2_abian(k1_zfmisc_1(A),k1_knaster(A,B),B) ) ),
    file(knaster,t6_knaster),
    []).

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),
    []).