% Mizar problem: t19_procal_1,procal_1,224,26 
fof(t19_procal_1,conjecture,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => r2_hidden(k12_cqc_lang(k11_cqc_lang(A,B),A),k4_cqc_the1) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,dt_k10_cqc_lang,dt_k11_cqc_lang,dt_k12_cqc_lang,dt_k12_qc_lang1,dt_k13_cqc_lang,dt_k13_qc_lang1,dt_k1_zfmisc_1,dt_k2_qc_lang2,dt_k3_qc_lang2,dt_k4_cqc_the1,dt_k7_cqc_lang,dt_k8_qc_lang1,dt_m1_subset_1,dt_m2_subset_1,existence_m1_subset_1,existence_m2_subset_1,fc2_cqc_lang,redefinition_k10_cqc_lang,redefinition_k11_cqc_lang,redefinition_k12_cqc_lang,redefinition_k13_cqc_lang,redefinition_m2_subset_1,t18_procal_1,t25_lukasi_1,t27_lukasi_1,t34_lukasi_1,t3_lukasi_1,t3_procal_1]),
    [file(procal_1,t19_procal_1)]).

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

fof(dt_k10_cqc_lang,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k7_cqc_lang)
     => m2_subset_1(k10_cqc_lang(A),k8_qc_lang1,k7_cqc_lang) ) ),
    file(cqc_lang,k10_cqc_lang),
    []).

fof(dt_k11_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => m2_subset_1(k11_cqc_lang(A,B),k8_qc_lang1,k7_cqc_lang) ) ),
    file(cqc_lang,k11_cqc_lang),
    []).

fof(dt_k12_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => m2_subset_1(k12_cqc_lang(A,B),k8_qc_lang1,k7_cqc_lang) ) ),
    file(cqc_lang,k12_cqc_lang),
    []).

fof(dt_k12_qc_lang1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k12_qc_lang1(A),k8_qc_lang1) ) ),
    file(qc_lang1,k12_qc_lang1),
    []).

fof(dt_k13_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => m2_subset_1(k13_cqc_lang(A,B),k8_qc_lang1,k7_cqc_lang) ) ),
    file(cqc_lang,k13_cqc_lang),
    []).

fof(dt_k13_qc_lang1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k8_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k13_qc_lang1(A,B),k8_qc_lang1) ) ),
    file(qc_lang1,k13_qc_lang1),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_qc_lang2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k8_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k2_qc_lang2(A,B),k8_qc_lang1) ) ),
    file(qc_lang2,k2_qc_lang2),
    []).

fof(dt_k3_qc_lang2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k8_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k3_qc_lang2(A,B),k8_qc_lang1) ) ),
    file(qc_lang2,k3_qc_lang2),
    []).

fof(dt_k4_cqc_the1,axiom,(
    m1_subset_1(k4_cqc_the1,k1_zfmisc_1(k7_cqc_lang)) ),
    file(cqc_the1,k4_cqc_the1),
    []).

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

fof(redefinition_k10_cqc_lang,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k7_cqc_lang)
     => k10_cqc_lang(A) = k12_qc_lang1(A) ) ),
    file(cqc_lang,k10_cqc_lang),
    []).

fof(redefinition_k11_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => k11_cqc_lang(A,B) = k13_qc_lang1(A,B) ) ),
    file(cqc_lang,k11_cqc_lang),
    []).

fof(redefinition_k12_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => k12_cqc_lang(A,B) = k2_qc_lang2(A,B) ) ),
    file(cqc_lang,k12_cqc_lang),
    []).

fof(redefinition_k13_cqc_lang,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k7_cqc_lang)
        & m1_subset_1(B,k7_cqc_lang) )
     => k13_cqc_lang(A,B) = k3_qc_lang2(A,B) ) ),
    file(cqc_lang,k13_cqc_lang),
    []).

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(t18_procal_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => r2_hidden(k12_cqc_lang(k13_cqc_lang(k10_cqc_lang(A),k10_cqc_lang(B)),k10_cqc_lang(k11_cqc_lang(A,B))),k4_cqc_the1) ) ) ),
    file(procal_1,t18_procal_1),
    []).

fof(t25_lukasi_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => r2_hidden(k12_cqc_lang(k10_cqc_lang(k10_cqc_lang(A)),A),k4_cqc_the1) ) ),
    file(lukasi_1,t25_lukasi_1),
    []).

fof(t27_lukasi_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => r2_hidden(k12_cqc_lang(A,k10_cqc_lang(k10_cqc_lang(A))),k4_cqc_the1) ) ),
    file(lukasi_1,t27_lukasi_1),
    []).

fof(t34_lukasi_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ( r2_hidden(k12_cqc_lang(A,B),k4_cqc_the1)
          <=> r2_hidden(k12_cqc_lang(k10_cqc_lang(B),k10_cqc_lang(A)),k4_cqc_the1) ) ) ) ),
    file(lukasi_1,t34_lukasi_1),
    []).

fof(t3_lukasi_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => ! [C] : 
              ( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
             => ( ( r2_hidden(k12_cqc_lang(A,B),k4_cqc_the1)
                  & r2_hidden(k12_cqc_lang(B,C),k4_cqc_the1) )
               => r2_hidden(k12_cqc_lang(A,C),k4_cqc_the1) ) ) ) ) ),
    file(lukasi_1,t3_lukasi_1),
    []).

fof(t3_procal_1,axiom,(
    ! [A] : 
      ( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
     => ! [B] : 
          ( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
         => r2_hidden(k12_cqc_lang(A,k13_cqc_lang(A,B)),k4_cqc_the1) ) ) ),
    file(procal_1,t3_procal_1),
    []).