% Mizar problem: t39_sublemma,sublemma,780,64 
fof(t39_sublemma,conjecture,(
    ! [A] : 
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] : 
          ( m2_subset_1(B,k16_substut1,k38_substut1)
         => ! [C] : 
              ( m1_substut1(C,k8_sublemma(B,A))
             => ( ( v3_substut1(k8_sublemma(B,A))
                  & r2_hidden(A,k2_relat_1(k7_substut1(A,k15_cqc_lang(A,k2_sublemma(B)),C))) )
               => ( ~ r2_hidden(k35_substut1(k32_substut1(k10_sublemma(k8_sublemma(B,A),C))),k2_relat_1(k7_substut1(A,k15_cqc_lang(A,k2_sublemma(B)),C)))
                  & ~ r2_hidden(k35_substut1(k32_substut1(k10_sublemma(k8_sublemma(B,A),C))),k9_substut1(k2_sublemma(B))) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_fraenkel,cc1_funct_1,cc1_nat_1,cc1_relat_1,cc1_relset_1,cc2_funct_1,cc2_nat_1,cc3_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k4_subset_1,connectedness_r1_xreal_0,d10_substut1,d11_substut1,d12_substut1,d35_substut1,d36_substut1,d4_cqc_lang,d4_xboole_0,d5_tarski,d6_sublemma,d8_cqc_sim1,d9_substut1,dt_k10_cqc_sim1,dt_k10_sublemma,dt_k10_substut1,dt_k11_substut1,dt_k12_substut1,dt_k13_substut1,dt_k14_qc_lang1,dt_k15_cqc_lang,dt_k16_substut1,dt_k18_substut1,dt_k19_substut1,dt_k1_mcart_1,dt_k1_numbers,dt_k1_qc_lang1,dt_k1_substut1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_qc_lang1,dt_k21_qc_lang1,dt_k22_substut1,dt_k23_substut1,dt_k24_substut1,dt_k2_mcart_1,dt_k2_qc_lang1,dt_k2_qc_lang3,dt_k2_relat_1,dt_k2_sublemma,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k32_substut1,dt_k33_substut1,dt_k34_substut1,dt_k35_substut1,dt_k38_substut1,dt_k3_qc_lang1,dt_k4_qc_lang1,dt_k4_subset_1,dt_k4_tarski,dt_k4_xboole_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_qc_lang3,dt_k6_qc_lang3,dt_k7_cqc_lang,dt_k7_substut1,dt_k8_qc_lang1,dt_k8_sublemma,dt_k9_sublemma,dt_k9_substut1,dt_m1_subset_1,dt_m1_substut1,dt_m2_subset_1,existence_m1_subset_1,existence_m1_substut1,existence_m2_subset_1,fc12_relat_1,fc1_ordinal2,fc1_qc_lang1,fc1_subset_1,fc1_substut1,fc2_cqc_lang,fc2_qc_lang1,fc2_relat_1,fc2_subset_1,fc2_substut1,fc3_qc_lang1,fc3_relat_1,fc3_subset_1,fc3_substut1,fc4_qc_lang1,fc4_relat_1,fc4_subset_1,fc4_substut1,fc6_relat_1,fc7_substut1,fc8_relat_1,fraenkel_a_0_0_cqc_lang,fraenkel_a_1_3_substut1,fraenkel_a_1_4_substut1,fraenkel_a_3_0_sublemma,fraenkel_a_3_1_sublemma,idempotence_k2_xboole_0,idempotence_k4_subset_1,rc1_fraenkel,rc1_funct_1,rc1_nat_1,rc1_relat_1,rc1_sublemma,rc1_subset_1,rc1_substut1,rc2_funct_1,rc2_nat_1,rc2_relat_1,rc2_subset_1,rc2_substut1,rc3_funct_1,rc3_nat_1,rc3_relat_1,rc4_funct_1,redefinition_k15_cqc_lang,redefinition_k18_substut1,redefinition_k19_substut1,redefinition_k22_substut1,redefinition_k23_substut1,redefinition_k2_sublemma,redefinition_k33_substut1,redefinition_k34_substut1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k8_sublemma,redefinition_k9_sublemma,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,t1_boole,t1_subset,t27_sublemma,t2_subset,t2_tarski,t3_boole,t3_subset,t4_boole,t4_subset,t5_subset,t6_boole,t7_boole,t7_mcart_1,t7_xboole_1,t8_boole,t8_qc_lang2]),
    [file(sublemma,t39_sublemma)]).

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

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_nat_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(nat_1,cc1_nat_1),
    []).

fof(cc1_relat_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_1),
    []).

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_nat_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(nat_1,cc2_nat_1),
    []).

fof(cc3_nat_1,axiom,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ( v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(nat_1,cc3_nat_1),
    []).

fof(commutativity_k2_tarski,axiom,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    []).

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_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(connectedness_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    []).

fof(d10_substut1,axiom,(
    ! [A] : 
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => k11_substut1(A) = a_1_4_substut1(A) ) ),
    file(substut1,d10_substut1),
    []).

fof(d11_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] : 
          ( ( v1_finset_1(B)
            & m1_subset_1(B,k1_substut1) )
         => k12_substut1(A,B) = k4_xboole_0(k5_numbers,k4_subset_1(k5_numbers,k10_substut1(A),k11_substut1(B))) ) ) ),
    file(substut1,d11_substut1),
    []).

fof(d12_substut1,axiom,(
    ! [A] : 
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => ! [B] : 
          ( m1_subset_1(B,k8_qc_lang1)
         => k13_substut1(A,B) = k10_cqc_sim1(k12_substut1(B,A)) ) ) ),
    file(substut1,d12_substut1),
    []).

fof(d35_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => k32_substut1(A) = A ) ),
    file(substut1,d35_substut1),
    []).

fof(d36_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => ( ( r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
         => k35_substut1(A) = k2_qc_lang3(k13_substut1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A)),k21_qc_lang1(k33_substut1(A)))) )
        & ( ~ r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
         => k35_substut1(A) = k20_qc_lang1(k33_substut1(A)) ) ) ) ),
    file(substut1,d36_substut1),
    []).

fof(d4_cqc_lang,axiom,(
    k7_cqc_lang = a_0_0_cqc_lang ),
    file(cqc_lang,d4_cqc_lang),
    []).

fof(d4_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    []).

fof(d5_tarski,axiom,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    []).

fof(d6_sublemma,axiom,(
    ! [A] : 
      ( ( v1_sublemma(A)
        & m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1)) )
     => ! [B] : 
          ( m1_substut1(B,A)
         => ( v3_substut1(A)
           => k10_sublemma(A,B) = k24_substut1(A,B) ) ) ) ),
    file(sublemma,d6_sublemma),
    []).

fof(d8_cqc_sim1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k10_cqc_sim1(A)
          <=> ( r2_hidden(B,A)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,A)
                   => r1_xreal_0(B,C) ) ) ) ) ) ) ),
    file(cqc_sim1,d8_cqc_sim1),
    []).

fof(d9_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => k10_substut1(A) = a_1_3_substut1(A) ) ),
    file(substut1,d9_substut1),
    []).

fof(dt_k10_cqc_sim1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => m2_subset_1(k10_cqc_sim1(A),k1_numbers,k5_numbers) ) ),
    file(cqc_sim1,k10_cqc_sim1),
    []).

fof(dt_k10_sublemma,axiom,(
    ! [A,B] : 
      ( ( v1_sublemma(A)
        & m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
        & m1_substut1(B,A) )
     => m2_subset_1(k10_sublemma(A,B),k16_substut1,k38_substut1) ) ),
    file(sublemma,k10_sublemma),
    []).

fof(dt_k10_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v1_finset_1(k10_substut1(A))
        & m1_subset_1(k10_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) ),
    file(substut1,k10_substut1),
    []).

fof(dt_k11_substut1,axiom,(
    ! [A] : 
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => ( v1_finset_1(k11_substut1(A))
        & m1_subset_1(k11_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) ),
    file(substut1,k11_substut1),
    []).

fof(dt_k12_substut1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k8_qc_lang1)
        & v1_finset_1(B)
        & m1_subset_1(B,k1_substut1) )
     => ( ~ v1_xboole_0(k12_substut1(A,B))
        & m1_subset_1(k12_substut1(A,B),k1_zfmisc_1(k5_numbers)) ) ) ),
    file(substut1,k12_substut1),
    []).

fof(dt_k13_substut1,axiom,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m2_subset_1(k13_substut1(A,B),k1_numbers,k5_numbers) ) ),
    file(substut1,k13_substut1),
    []).

fof(dt_k14_qc_lang1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k14_qc_lang1(A,B),k8_qc_lang1) ) ),
    file(qc_lang1,k14_qc_lang1),
    []).

fof(dt_k15_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k7_cqc_lang) )
     => m2_subset_1(k15_cqc_lang(A,B),k8_qc_lang1,k7_cqc_lang) ) ),
    file(cqc_lang,k15_cqc_lang),
    []).

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

fof(dt_k18_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k18_substut1(A),k8_qc_lang1) ) ),
    file(substut1,k18_substut1),
    []).

fof(dt_k19_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k19_substut1(A),k1_substut1) ) ),
    file(substut1,k19_substut1),
    []).

fof(dt_k1_mcart_1,axiom,(
    $true ),
    file(mcart_1,k1_mcart_1),
    []).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    []).

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

fof(dt_k1_substut1,axiom,(
    $true ),
    file(substut1,k1_substut1),
    []).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

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_k20_qc_lang1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m2_subset_1(k20_qc_lang1(A),k1_qc_lang1,k2_qc_lang1) ) ),
    file(qc_lang1,k20_qc_lang1),
    []).

fof(dt_k21_qc_lang1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k21_qc_lang1(A),k8_qc_lang1) ) ),
    file(qc_lang1,k21_qc_lang1),
    []).

fof(dt_k22_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => m1_subset_1(k22_substut1(A),k16_substut1) ) ),
    file(substut1,k22_substut1),
    []).

fof(dt_k23_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => m2_subset_1(k23_substut1(A),k1_qc_lang1,k2_qc_lang1) ) ),
    file(substut1,k23_substut1),
    []).

fof(dt_k24_substut1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
        & m1_substut1(B,A) )
     => m1_subset_1(k24_substut1(A,B),k16_substut1) ) ),
    file(substut1,k24_substut1),
    []).

fof(dt_k2_mcart_1,axiom,(
    $true ),
    file(mcart_1,k2_mcart_1),
    []).

fof(dt_k2_qc_lang1,axiom,(
    m1_subset_1(k2_qc_lang1,k1_zfmisc_1(k1_qc_lang1)) ),
    file(qc_lang1,k2_qc_lang1),
    []).

fof(dt_k2_qc_lang3,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k2_qc_lang3(A),k1_qc_lang1,k2_qc_lang1) ) ),
    file(qc_lang3,k2_qc_lang3),
    []).

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    []).

fof(dt_k2_sublemma,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k38_substut1)
     => m2_subset_1(k2_sublemma(A),k8_qc_lang1,k7_cqc_lang) ) ),
    file(sublemma,k2_sublemma),
    []).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    []).

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_k32_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k32_substut1(A),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ),
    file(substut1,k32_substut1),
    []).

fof(dt_k33_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m1_subset_1(k33_substut1(A),k8_qc_lang1) ) ),
    file(substut1,k33_substut1),
    []).

fof(dt_k34_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m1_subset_1(k34_substut1(A),k1_substut1) ) ),
    file(substut1,k34_substut1),
    []).

fof(dt_k35_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m2_subset_1(k35_substut1(A),k1_qc_lang1,k2_qc_lang1) ) ),
    file(substut1,k35_substut1),
    []).

fof(dt_k38_substut1,axiom,(
    m1_subset_1(k38_substut1,k1_zfmisc_1(k16_substut1)) ),
    file(substut1,k38_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_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    []).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    []).

fof(dt_k5_numbers,axiom,(
    m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ),
    file(numbers,k5_numbers),
    []).

fof(dt_k5_ordinal2,axiom,(
    $true ),
    file(ordinal2,k5_ordinal2),
    []).

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_k7_substut1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1)
        & m1_subset_1(C,k1_substut1) )
     => ( v1_finset_1(k7_substut1(A,B,C))
        & m1_subset_1(k7_substut1(A,B,C),k1_substut1) ) ) ),
    file(substut1,k7_substut1),
    []).

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

fof(dt_k8_sublemma,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k38_substut1)
        & m1_subset_1(B,k2_qc_lang1) )
     => ( v1_sublemma(k8_sublemma(A,B))
        & m1_subset_1(k8_sublemma(A,B),k2_zfmisc_1(k16_substut1,k2_qc_lang1)) ) ) ),
    file(sublemma,k8_sublemma),
    []).

fof(dt_k9_sublemma,axiom,(
    ! [A] : 
      ( ( v1_sublemma(A)
        & m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1)) )
     => m2_subset_1(k9_sublemma(A),k16_substut1,k38_substut1) ) ),
    file(sublemma,k9_sublemma),
    []).

fof(dt_k9_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k9_substut1(A),k1_zfmisc_1(k2_qc_lang1)) ) ),
    file(substut1,k9_substut1),
    []).

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

fof(dt_m1_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] : 
          ( m1_substut1(B,A)
         => m1_subset_1(B,k1_substut1) ) ) ),
    file(substut1,m1_substut1),
    []).

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_m1_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ? [B] : m1_substut1(B,A) ) ),
    file(substut1,m1_substut1),
    []).

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_ordinal2,axiom,
    ( v1_ordinal1(k5_ordinal2)
    & v2_ordinal1(k5_ordinal2)
    & v3_ordinal1(k5_ordinal2)
    & ~ v1_xboole_0(k5_ordinal2) ),
    file(ordinal2,fc1_ordinal2),
    []).

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(fc1_substut1,axiom,(
    ~ v1_xboole_0(k1_substut1) ),
    file(substut1,fc1_substut1),
    []).

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

fof(fc2_qc_lang1,axiom,(
    ~ v1_xboole_0(k2_qc_lang1) ),
    file(qc_lang1,fc2_qc_lang1),
    []).

fof(fc2_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k2_xboole_0(A,B)) ) ),
    file(relat_1,fc2_relat_1),
    []).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    []).

fof(fc2_substut1,axiom,
    ( ~ v1_xboole_0(k1_substut1)
    & v1_fraenkel(k1_substut1) ),
    file(substut1,fc2_substut1),
    []).

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

fof(fc3_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k4_xboole_0(A,B)) ) ),
    file(relat_1,fc3_relat_1),
    []).

fof(fc3_subset_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    []).

fof(fc3_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => v1_finset_1(k9_substut1(A)) ) ),
    file(substut1,fc3_substut1),
    []).

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

fof(fc6_relat_1,axiom,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A) )
     => ~ v1_xboole_0(k2_relat_1(A)) ) ),
    file(relat_1,fc6_relat_1),
    []).

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

fof(fc8_relat_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_xboole_0(k2_relat_1(A))
        & v1_relat_1(k2_relat_1(A)) ) ) ),
    file(relat_1,fc8_relat_1),
    []).

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(fraenkel_a_1_3_substut1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k8_qc_lang1)
     => ( r2_hidden(A,a_1_3_substut1(B))
      <=> ? [C] : 
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & r2_hidden(k2_qc_lang3(C),k9_substut1(B)) ) ) ) ),
    file(substut1,a_1_3_substut1),
    []).

fof(fraenkel_a_1_4_substut1,axiom,(
    ! [A,B] : 
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_substut1) )
     => ( r2_hidden(A,a_1_4_substut1(B))
      <=> ? [C] : 
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & r2_hidden(k2_qc_lang3(C),k2_relat_1(B)) ) ) ) ),
    file(substut1,a_1_4_substut1),
    []).

fof(fraenkel_a_3_0_sublemma,axiom,(
    ! [A,B,C,D] : 
      ( ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
        & m2_subset_1(C,k16_substut1,k38_substut1)
        & m1_substut1(D,k8_sublemma(C,B)) )
     => ( r2_hidden(A,a_3_0_sublemma(B,C,D))
      <=> ? [E] : 
            ( m2_subset_1(E,k1_numbers,k5_numbers)
            & A = E
            & r2_hidden(k2_qc_lang3(E),k2_relat_1(k7_substut1(k20_qc_lang1(k2_sublemma(k10_sublemma(k8_sublemma(C,B),D))),k2_sublemma(k10_sublemma(k8_sublemma(C,B),D)),k19_substut1(k10_sublemma(k8_sublemma(C,B),D))))) ) ) ) ),
    file(sublemma,a_3_0_sublemma),
    []).

fof(fraenkel_a_3_1_sublemma,axiom,(
    ! [A,B,C,D] : 
      ( ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
        & m2_subset_1(C,k16_substut1,k38_substut1)
        & m1_substut1(D,k8_sublemma(C,B)) )
     => ( r2_hidden(A,a_3_1_sublemma(B,C,D))
      <=> ? [E] : 
            ( m2_subset_1(E,k1_numbers,k5_numbers)
            & A = E
            & r2_hidden(k2_qc_lang3(E),k9_substut1(k21_qc_lang1(k2_sublemma(k10_sublemma(k8_sublemma(C,B),D))))) ) ) ) ),
    file(sublemma,a_3_1_sublemma),
    []).

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

fof(idempotence_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,B) = B ) ),
    file(subset_1,k4_subset_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_nat_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(nat_1,rc1_nat_1),
    []).

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

fof(rc1_sublemma,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
      & v1_sublemma(A) ) ),
    file(sublemma,rc1_sublemma),
    []).

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_substut1,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k1_substut1)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A) ) ),
    file(substut1,rc1_substut1),
    []).

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_nat_1,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v3_ordinal1(A) ) ),
    file(nat_1,rc2_nat_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_nat_1,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(nat_1,rc3_nat_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_k15_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k7_cqc_lang) )
     => k15_cqc_lang(A,B) = k14_qc_lang1(A,B) ) ),
    file(cqc_lang,k15_cqc_lang),
    []).

fof(redefinition_k18_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => k18_substut1(A) = k1_mcart_1(A) ) ),
    file(substut1,k18_substut1),
    []).

fof(redefinition_k19_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k16_substut1)
     => k19_substut1(A) = k2_mcart_1(A) ) ),
    file(substut1,k19_substut1),
    []).

fof(redefinition_k22_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => k22_substut1(A) = k1_mcart_1(A) ) ),
    file(substut1,k22_substut1),
    []).

fof(redefinition_k23_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => k23_substut1(A) = k2_mcart_1(A) ) ),
    file(substut1,k23_substut1),
    []).

fof(redefinition_k2_sublemma,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k38_substut1)
     => k2_sublemma(A) = k1_mcart_1(A) ) ),
    file(sublemma,k2_sublemma),
    []).

fof(redefinition_k33_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => k33_substut1(A) = k1_mcart_1(A) ) ),
    file(substut1,k33_substut1),
    []).

fof(redefinition_k34_substut1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => k34_substut1(A) = k2_mcart_1(A) ) ),
    file(substut1,k34_substut1),
    []).

fof(redefinition_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(redefinition_k5_numbers,axiom,(
    k5_numbers = k5_ordinal2 ),
    file(numbers,k5_numbers),
    []).

fof(redefinition_k8_sublemma,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k38_substut1)
        & m1_subset_1(B,k2_qc_lang1) )
     => k8_sublemma(A,B) = k4_tarski(A,B) ) ),
    file(sublemma,k8_sublemma),
    []).

fof(redefinition_k9_sublemma,axiom,(
    ! [A] : 
      ( ( v1_sublemma(A)
        & m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1)) )
     => k9_sublemma(A) = k1_mcart_1(A) ) ),
    file(sublemma,k9_sublemma),
    []).

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(reflexivity_r1_xreal_0,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    []).

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(t27_sublemma,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] : 
          ( m1_substut1(B,A)
         => ( v3_substut1(A)
           => ( k18_substut1(k24_substut1(A,B)) = k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A)))
              & k19_substut1(k24_substut1(A,B)) = B ) ) ) ) ),
    file(sublemma,t27_sublemma),
    []).

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(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(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(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t7_mcart_1,axiom,(
    ! [A,B] : 
      ( k1_mcart_1(k4_tarski(A,B)) = A
      & k2_mcart_1(k4_tarski(A,B)) = B ) ),
    file(mcart_1,t7_mcart_1),
    []).

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_qc_lang2,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] : 
          ( m1_subset_1(B,k8_qc_lang1)
         => ( k20_qc_lang1(k14_qc_lang1(A,B)) = A
            & k21_qc_lang1(k14_qc_lang1(A,B)) = B ) ) ) ),
    file(qc_lang2,t8_qc_lang2),
    []).