% Mizar problem: t97_qc_lang2,qc_lang2,1183,25 
fof(t97_qc_lang2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
       => ( r3_qc_lang2(k18_qc_lang1(A),A)
          & r3_qc_lang2(k19_qc_lang1(A),A) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finseq_1,dt_k18_qc_lang1,dt_k19_qc_lang1,dt_k1_xboole_0,dt_k8_qc_lang1,dt_m1_subset_1,existence_m1_subset_1,fc2_finseq_1,irreflexivity_r3_qc_lang2,rc1_finseq_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_subset,t2_subset,t66_qc_lang2,t6_boole,t73_qc_lang2,t7_boole,t8_boole]),
    [file(qc_lang2,t97_qc_lang2)]).

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(dt_k18_qc_lang1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k18_qc_lang1(A),k8_qc_lang1) ) ),
    file(qc_lang1,k18_qc_lang1),
    []).

fof(dt_k19_qc_lang1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k19_qc_lang1(A),k8_qc_lang1) ) ),
    file(qc_lang1,k19_qc_lang1),
    []).

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

fof(dt_k8_qc_lang1,axiom,(
    ~ v1_xboole_0(k8_qc_lang1) ),
    file(qc_lang1,k8_qc_lang1),
    []).

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(irreflexivity_r3_qc_lang2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k8_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => ~ r3_qc_lang2(A,A) ) ),
    file(qc_lang2,r3_qc_lang2),
    []).

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(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_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_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(t66_qc_lang2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] : 
          ( m1_subset_1(B,k8_qc_lang1)
         => ( v4_qc_lang1(A)
           => ( r1_qc_lang2(B,A)
            <=> ( B = k18_qc_lang1(A)
                | B = k19_qc_lang1(A) ) ) ) ) ) ),
    file(qc_lang2,t66_qc_lang2),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t73_qc_lang2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] : 
          ( m1_subset_1(B,k8_qc_lang1)
         => ( r1_qc_lang2(A,B)
           => r3_qc_lang2(A,B) ) ) ) ),
    file(qc_lang2,t73_qc_lang2),
    []).

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),
    []).