% Mizar problem: t29_sublemma,sublemma,640,71 
fof(t29_sublemma,conjecture,(
    ! [A] : 
      ( m2_subset_1(A,k16_substut1,k38_substut1)
     => ( v7_substut1(A)
       => k39_substut1(A) = k12_sublemma(A,k39_substut1(k11_sublemma(A))) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,cc1_relat_1,cc2_funct_1,d4_cqc_lang,d7_sublemma,d8_sublemma,dt_k11_sublemma,dt_k12_sublemma,dt_k16_substut1,dt_k1_qc_lang1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k31_substut1,dt_k36_substut1,dt_k37_substut1,dt_k38_substut1,dt_k39_substut1,dt_k3_qc_lang1,dt_k4_qc_lang1,dt_k5_qc_lang3,dt_k6_qc_lang3,dt_k7_cqc_lang,dt_k8_qc_lang1,dt_m1_subset_1,dt_m2_subset_1,existence_m1_subset_1,existence_m2_subset_1,fc12_relat_1,fc1_qc_lang1,fc1_subset_1,fc2_cqc_lang,fc3_qc_lang1,fc4_qc_lang1,fc4_relat_1,fc4_substut1,fc7_substut1,fraenkel_a_0_0_cqc_lang,rc1_funct_1,rc1_relat_1,rc1_subset_1,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_substut1,rc3_funct_1,rc3_relat_1,rc4_funct_1,redefinition_k39_substut1,redefinition_m2_subset_1,reflexivity_r1_tarski,t1_subset,t2_subset,t2_tarski,t32_substut1,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(sublemma,t29_sublemma)]).

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_relat_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_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(d4_cqc_lang,axiom,(
    k7_cqc_lang = a_0_0_cqc_lang ),
    file(cqc_lang,d4_cqc_lang),
    []).

fof(d7_sublemma,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k16_substut1,k38_substut1)
     => ( v7_substut1(A)
       => k11_sublemma(A) = k31_substut1(A) ) ) ),
    file(sublemma,d7_sublemma),
    []).

fof(d8_sublemma,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k16_substut1,k38_substut1)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ( ( v7_substut1(A)
              & B = k39_substut1(k11_sublemma(A)) )
           => k12_sublemma(A,B) = k36_substut1(A,B) ) ) ) ),
    file(sublemma,d8_sublemma),
    []).

fof(dt_k11_sublemma,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k38_substut1)
     => m2_subset_1(k11_sublemma(A),k16_substut1,k38_substut1) ) ),
    file(sublemma,k11_sublemma),
    []).

fof(dt_k12_sublemma,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k38_substut1)
        & m1_subset_1(B,k7_cqc_lang) )
     => m2_subset_1(k12_sublemma(A,B),k8_qc_lang1,k7_cqc_lang) ) ),
    file(sublemma,k12_sublemma),
    []).

fof(dt_k16_substut1,axiom,(
    ~ v1_xboole_0(k16_substut1) ),
    file(substut1,k16_substut1),
    []).

fof(dt_k1_qc_lang1,axiom,(
    $true ),
    file(qc_lang1,k1_qc_lang1),
    []).

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_k31_substut1,axiom,(
    ! [A] : m1_subset_1(k31_substut1(A),k16_substut1) ),
    file(substut1,k31_substut1),
    []).

fof(dt_k36_substut1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k16_substut1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k36_substut1(A,B),k8_qc_lang1) ) ),
    file(substut1,k36_substut1),
    []).

fof(dt_k37_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k37_substut1(A),k8_qc_lang1) ) ),
    file(substut1,k37_substut1),
    []).

fof(dt_k38_substut1,axiom,(
    m1_subset_1(k38_substut1,k1_zfmisc_1(k16_substut1)) ),
    file(substut1,k38_substut1),
    []).

fof(dt_k39_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k38_substut1)
     => m2_subset_1(k39_substut1(A),k8_qc_lang1,k7_cqc_lang) ) ),
    file(substut1,k39_substut1),
    []).

fof(dt_k3_qc_lang1,axiom,(
    m1_subset_1(k3_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ),
    file(qc_lang1,k3_qc_lang1),
    []).

fof(dt_k4_qc_lang1,axiom,(
    m1_subset_1(k4_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ),
    file(qc_lang1,k4_qc_lang1),
    []).

fof(dt_k5_qc_lang3,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k5_qc_lang3(A),k1_zfmisc_1(k4_qc_lang1)) ) ),
    file(qc_lang3,k5_qc_lang3),
    []).

fof(dt_k6_qc_lang3,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k6_qc_lang3(A),k1_zfmisc_1(k3_qc_lang1)) ) ),
    file(qc_lang3,k6_qc_lang3),
    []).

fof(dt_k7_cqc_lang,axiom,(
    m1_subset_1(k7_cqc_lang,k1_zfmisc_1(k8_qc_lang1)) ),
    file(cqc_lang,k7_cqc_lang),
    []).

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(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(fc12_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ),
    file(relat_1,fc12_relat_1),
    []).

fof(fc1_qc_lang1,axiom,(
    ~ v1_xboole_0(k1_qc_lang1) ),
    file(qc_lang1,fc1_qc_lang1),
    []).

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

fof(fc2_cqc_lang,axiom,(
    ~ v1_xboole_0(k7_cqc_lang) ),
    file(cqc_lang,fc2_cqc_lang),
    []).

fof(fc3_qc_lang1,axiom,(
    ~ v1_xboole_0(k3_qc_lang1) ),
    file(qc_lang1,fc3_qc_lang1),
    []).

fof(fc4_qc_lang1,axiom,(
    ~ v1_xboole_0(k4_qc_lang1) ),
    file(qc_lang1,fc4_qc_lang1),
    []).

fof(fc4_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ),
    file(relat_1,fc4_relat_1),
    []).

fof(fc4_substut1,axiom,
    ( ~ v1_xboole_0(k16_substut1)
    & v1_substut1(k16_substut1) ),
    file(substut1,fc4_substut1),
    []).

fof(fc7_substut1,axiom,(
    ~ v1_xboole_0(k38_substut1) ),
    file(substut1,fc7_substut1),
    []).

fof(fraenkel_a_0_0_cqc_lang,axiom,(
    ! [A] : 
      ( r2_hidden(A,a_0_0_cqc_lang)
    <=> ? [B] : 
          ( m1_subset_1(B,k8_qc_lang1)
          & A = B
          & k6_qc_lang3(B) = k1_xboole_0
          & k5_qc_lang3(B) = k1_xboole_0 ) ) ),
    file(cqc_lang,a_0_0_cqc_lang),
    []).

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

fof(rc1_relat_1,axiom,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc1_relat_1),
    []).

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_relat_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc2_relat_1),
    []).

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_substut1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_substut1(A) ) ),
    file(substut1,rc2_substut1),
    []).

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_relat_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ),
    file(relat_1,rc3_relat_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(redefinition_k39_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k38_substut1)
     => k39_substut1(A) = k37_substut1(A) ) ),
    file(substut1,k39_substut1),
    []).

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

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(t32_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => ( v7_substut1(A)
       => k37_substut1(A) = k36_substut1(A,k37_substut1(k31_substut1(A))) ) ) ),
    file(substut1,t32_substut1),
    []).

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