% Mizar problem: t19_algspec1,algspec1,309,54 
fof(t19_algspec1,conjecture,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => r6_pboole(k2_xboole_0(A,B),k1_algspec1(k2_xboole_0(A,B),C),k1_circcomb(A,B,k1_algspec1(A,C),k1_algspec1(B,C))) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_relset_1,commutativity_k2_xboole_0,commutativity_k3_xboole_0,d1_algspec1,dt_k1_algspec1,dt_k1_circcomb,dt_k1_funct_4,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_xboole_0,dt_k4_relset_1,dt_k6_partfun1,dt_k6_relat_1,dt_k7_relat_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_relat_1,fc2_funct_1,fc2_relat_1,fc4_funct_1,idempotence_k1_circcomb,idempotence_k1_funct_4,idempotence_k2_xboole_0,idempotence_k3_xboole_0,rc1_funct_1,redefinition_k1_circcomb,redefinition_k4_relset_1,redefinition_k6_partfun1,redefinition_m2_relset_1,redefinition_r6_pboole,reflexivity_r1_partfun1,reflexivity_r1_tarski,reflexivity_r6_pboole,symmetry_r1_partfun1,symmetry_r6_pboole,t107_relat_1,t131_partfun1,t15_funct_4,t17_xboole_1,t1_algspec1,t23_funct_4,t27_xboole_1,t31_funct_4,t3_subset,t88_relat_1,t90_relat_1]),
    [file(algspec1,t19_algspec1)]).

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(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_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    []).

fof(d1_algspec1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_algspec1(A,B) = k1_funct_4(k6_partfun1(A),k7_relat_1(B,A)) ) ),
    file(algspec1,d1_algspec1),
    []).

fof(dt_k1_algspec1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => m1_pboole(k1_algspec1(A,B),A) ) ),
    file(algspec1,k1_algspec1),
    []).

fof(dt_k1_circcomb,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_pboole(C,A)
        & m1_pboole(D,B) )
     => m1_pboole(k1_circcomb(A,B,C,D),k2_xboole_0(A,B)) ) ),
    file(circcomb,k1_circcomb),
    []).

fof(dt_k1_funct_4,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k1_funct_4(A,B))
        & v1_funct_1(k1_funct_4(A,B)) ) ) ),
    file(funct_4,k1_funct_4),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

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

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_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    []).

fof(dt_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(relset_1,k4_relset_1),
    []).

fof(dt_k6_partfun1,axiom,(
    ! [A] : 
      ( v1_partfun1(k6_partfun1(A),A,A)
      & m2_relset_1(k6_partfun1(A),A,A) ) ),
    file(partfun1,k6_partfun1),
    []).

fof(dt_k6_relat_1,axiom,(
    ! [A] : v1_relat_1(k6_relat_1(A)) ),
    file(relat_1,k6_relat_1),
    []).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    []).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    []).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

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

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(existence_m1_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    []).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

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

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc1_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k3_xboole_0(A,B)) ) ),
    file(relat_1,fc1_relat_1),
    []).

fof(fc2_funct_1,axiom,(
    ! [A] : 
      ( v1_relat_1(k6_relat_1(A))
      & v1_funct_1(k6_relat_1(A)) ) ),
    file(funct_1,fc2_funct_1),
    []).

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(fc4_funct_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) ),
    file(funct_1,fc4_funct_1),
    []).

fof(idempotence_k1_circcomb,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_pboole(C,A)
        & m1_pboole(D,B) )
     => k1_circcomb(A,B,C,C) = C ) ),
    file(circcomb,k1_circcomb),
    []).

fof(idempotence_k1_funct_4,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_funct_4(A,A) = A ) ),
    file(funct_4,k1_funct_4),
    []).

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

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

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

fof(redefinition_k1_circcomb,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_pboole(C,A)
        & m1_pboole(D,B) )
     => k1_circcomb(A,B,C,D) = k1_funct_4(C,D) ) ),
    file(circcomb,k1_circcomb),
    []).

fof(redefinition_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) ),
    file(relset_1,k4_relset_1),
    []).

fof(redefinition_k6_partfun1,axiom,(
    ! [A] : k6_partfun1(A) = k6_relat_1(A) ),
    file(partfun1,k6_partfun1),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(redefinition_r6_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ( r6_pboole(A,B,C)
      <=> B = C ) ) ),
    file(pboole,r6_pboole),
    []).

fof(reflexivity_r1_partfun1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_partfun1(A,A) ) ),
    file(partfun1,r1_partfun1),
    []).

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

fof(reflexivity_r6_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => r6_pboole(A,B,B) ) ),
    file(pboole,r6_pboole),
    []).

fof(symmetry_r1_partfun1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_partfun1(A,B)
       => r1_partfun1(B,A) ) ) ),
    file(partfun1,r1_partfun1),
    []).

fof(symmetry_r6_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ( r6_pboole(A,B,C)
       => r6_pboole(A,C,B) ) ) ),
    file(pboole,r6_pboole),
    []).

fof(t107_relat_1,axiom,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => k7_relat_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k7_relat_1(C,A),k7_relat_1(C,B)) ) ),
    file(relat_1,t107_relat_1),
    []).

fof(t131_partfun1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> ? [C] : 
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & r1_tarski(A,C)
                & r1_tarski(B,C) ) ) ) ) ),
    file(partfun1,t131_partfun1),
    []).

fof(t15_funct_4,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,k1_funct_4(B,C)) ) ) ) ),
    file(funct_4,t15_funct_4),
    []).

fof(t17_xboole_1,axiom,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ),
    file(xboole_1,t17_xboole_1),
    []).

fof(t1_algspec1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( r1_tarski(k3_xboole_0(k1_relat_1(A),k1_relat_1(B)),k1_relat_1(C))
               => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(k1_funct_4(B,A),C) ) ) ) ) ),
    file(algspec1,t1_algspec1),
    []).

fof(t23_funct_4,axiom,(
    ! [A,B] : k1_funct_4(k6_partfun1(A),k6_partfun1(B)) = k6_partfun1(k2_xboole_0(A,B)) ),
    file(funct_4,t23_funct_4),
    []).

fof(t27_xboole_1,axiom,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,D)) ) ),
    file(xboole_1,t27_xboole_1),
    []).

fof(t31_funct_4,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) ),
    file(funct_4,t31_funct_4),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t88_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k7_relat_1(B,A),B) ) ),
    file(relat_1,t88_relat_1),
    []).

fof(t90_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ),
    file(relat_1,t90_relat_1),
    []).