% Mizar problem: t87_cohsp_1,cohsp_1,3257,60 
fof(t87_cohsp_1,conjecture,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_classes1(B)
            & v1_coh_sp(B) )
         => ! [C,D] : 
              ( r2_hidden(k14_cohsp_1(C,D),k16_cohsp_1(A,B))
            <=> ( ( r2_hidden(C,A)
                  & D = k1_xboole_0 )
                | ( C = k1_xboole_0
                  & r2_hidden(D,B) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finseq_1,cc1_finset_1,cc3_cohsp_1,cc4_cohsp_1,commutativity_k2_xboole_0,d23_cohsp_1,d25_cohsp_1,d2_xboole_0,dt_k10_finseq_1,dt_k14_cohsp_1,dt_k16_cohsp_1,dt_k1_xboole_0,dt_k2_card_3,dt_k2_xboole_0,dt_k3_card_3,dt_m1_subset_1,existence_m1_subset_1,fc2_finseq_1,fc5_finseq_1,fc7_finseq_1,fc9_finset_1,fraenkel_a_1_3_cohsp_1,fraenkel_a_1_4_cohsp_1,idempotence_k2_xboole_0,rc1_cohsp_1,rc1_finseq_1,rc1_finset_1,rc1_partfun1,rc2_cohsp_1,rc3_finseq_1,rc6_cohsp_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_boole,t1_subset,t2_subset,t2_tarski,t6_boole,t7_boole,t81_cohsp_1,t8_boole]),
    [file(cohsp_1,t87_cohsp_1)]).

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_finset_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    []).

fof(cc3_cohsp_1,axiom,(
    ! [A] : 
      ( v1_classes1(A)
     => v2_finsub_1(A) ) ),
    file(cohsp_1,cc3_cohsp_1),
    []).

fof(cc4_cohsp_1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ( ~ v1_xboole_0(A)
        & v2_finsub_1(A)
        & v1_classes1(A)
        & v1_coh_sp(A)
        & v3_cohsp_1(A) ) ) ),
    file(cohsp_1,cc4_cohsp_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(d23_cohsp_1,axiom,(
    ! [A,B] : k14_cohsp_1(A,B) = k3_card_3(k2_card_3(k10_finseq_1(A,B))) ),
    file(cohsp_1,d23_cohsp_1),
    []).

fof(d25_cohsp_1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_classes1(B)
            & v1_coh_sp(B) )
         => k16_cohsp_1(A,B) = k2_xboole_0(a_1_3_cohsp_1(A),a_1_4_cohsp_1(B)) ) ) ),
    file(cohsp_1,d25_cohsp_1),
    []).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(dt_k10_finseq_1,axiom,(
    $true ),
    file(finseq_1,k10_finseq_1),
    []).

fof(dt_k14_cohsp_1,axiom,(
    $true ),
    file(cohsp_1,k14_cohsp_1),
    []).

fof(dt_k16_cohsp_1,axiom,(
    $true ),
    file(cohsp_1,k16_cohsp_1),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k2_card_3,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_card_3(A))
        & v1_funct_1(k2_card_3(A)) ) ) ),
    file(card_3,k2_card_3),
    []).

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

fof(dt_k3_card_3,axiom,(
    $true ),
    file(card_3,k3_card_3),
    []).

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

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

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(fc5_finseq_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(k10_finseq_1(A,B))
      & v1_funct_1(k10_finseq_1(A,B)) ) ),
    file(finseq_1,fc5_finseq_1),
    []).

fof(fc7_finseq_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(k10_finseq_1(A,B))
      & v1_funct_1(k10_finseq_1(A,B))
      & v1_finset_1(k10_finseq_1(A,B))
      & v1_finseq_1(k10_finseq_1(A,B)) ) ),
    file(finseq_1,fc7_finseq_1),
    []).

fof(fc9_finset_1,axiom,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_xboole_0(A,B)) ) ),
    file(finset_1,fc9_finset_1),
    []).

fof(fraenkel_a_1_3_cohsp_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( r2_hidden(A,a_1_3_cohsp_1(B))
      <=> ? [C] : 
            ( m1_subset_1(C,B)
            & A = k14_cohsp_1(C,k1_xboole_0) ) ) ) ),
    file(cohsp_1,a_1_3_cohsp_1),
    []).

fof(fraenkel_a_1_4_cohsp_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & v1_classes1(B)
        & v1_coh_sp(B) )
     => ( r2_hidden(A,a_1_4_cohsp_1(B))
      <=> ? [C] : 
            ( m1_subset_1(C,B)
            & A = k14_cohsp_1(k1_xboole_0,C) ) ) ) ),
    file(cohsp_1,a_1_4_cohsp_1),
    []).

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

fof(rc1_cohsp_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_classes1(A)
      & v1_finset_1(A)
      & v1_coh_sp(A) ) ),
    file(cohsp_1,rc1_cohsp_1),
    []).

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_finset_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    []).

fof(rc1_partfun1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    []).

fof(rc2_cohsp_1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes1(A)
        & v1_coh_sp(A) )
     => ? [B] : 
          ( m1_subset_1(B,A)
          & v1_finset_1(B) ) ) ),
    file(cohsp_1,rc2_cohsp_1),
    []).

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(rc6_cohsp_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v2_finsub_1(A)
      & v1_classes1(A)
      & v1_coh_sp(A)
      & v3_cohsp_1(A) ) ),
    file(cohsp_1,rc6_cohsp_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(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(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

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(t81_cohsp_1,axiom,(
    ! [A,B,C,D] : 
      ( k14_cohsp_1(A,B) = k14_cohsp_1(C,D)
    <=> ( A = C
        & B = D ) ) ),
    file(cohsp_1,t81_cohsp_1),
    []).

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