% Mizar problem: t13_closure1,closure1,571,35 
fof(t13_closure1,conjecture,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v2_closure1(k2_msualg_3(A,k7_mssubfam(A,B)),A,B)
        & m3_pboole(k2_msualg_3(A,k7_mssubfam(A,B)),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc3_mssubfam,cc3_pboole,cc4_mssubfam,cc5_mssubfam,cc6_mssubfam,cc7_mssubfam,cc8_mssubfam,d3_closure1,dt_k15_pralg_1,dt_k1_mboolean,dt_k2_closure1,dt_k2_msualg_3,dt_k3_closure1,dt_k7_mssubfam,dt_m1_pboole,dt_m2_pboole,dt_m3_pboole,dt_m4_pboole,existence_m1_pboole,existence_m2_pboole,existence_m3_pboole,existence_m4_pboole,rc1_funcop_1,rc1_funct_1,rc2_mssubfam,rc3_pboole,rc4_pboole,rc5_funct_1,rc5_mssubfam,rc5_pboole,redefinition_k2_closure1,redefinition_k3_closure1,redefinition_k7_mssubfam,reflexivity_r2_pboole,t9_closure1]),
    [file(closure1,t13_closure1)]).

fof(cc3_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v2_mssubfam(C,A,B)
           => v1_mssubfam(C,A,B) ) ) ) ),
    file(mssubfam,cc3_mssubfam),
    []).

fof(cc3_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ! [D] : 
          ( m3_pboole(D,A,B,C)
         => v1_funcop_1(D) ) ) ),
    file(pboole,cc3_pboole),
    []).

fof(cc4_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v4_mssubfam(C,A,B)
           => v3_mssubfam(C,A,B) ) ) ) ),
    file(mssubfam,cc4_mssubfam),
    []).

fof(cc5_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v4_mssubfam(C,A,B)
           => v5_mssubfam(C,A,B) ) ) ) ),
    file(mssubfam,cc5_mssubfam),
    []).

fof(cc6_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v5_mssubfam(C,A,B)
           => v2_relat_1(C) ) ) ) ),
    file(mssubfam,cc6_mssubfam),
    []).

fof(cc7_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v2_mssubfam(C,A,B)
           => v6_mssubfam(C,A,B) ) ) ) ),
    file(mssubfam,cc7_mssubfam),
    []).

fof(cc8_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
         => ( v6_mssubfam(C,A,B)
           => v2_relat_1(C) ) ) ) ),
    file(mssubfam,cc8_mssubfam),
    []).

fof(d3_closure1,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
         => ( v2_closure1(C,A,B)
          <=> ! [D] : 
                ( m2_pboole(D,A,k7_mssubfam(A,B))
               => ! [E] : 
                    ( m2_pboole(E,A,k7_mssubfam(A,B))
                   => ( r2_pboole(A,D,E)
                     => r2_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,E)) ) ) ) ) ) ) ),
    file(closure1,d3_closure1),
    []).

fof(dt_k15_pralg_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k15_pralg_1(A,B))
        & v1_funct_1(k15_pralg_1(A,B)) ) ) ),
    file(pralg_1,k15_pralg_1),
    []).

fof(dt_k1_mboolean,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => m1_pboole(k1_mboolean(A,B),A) ) ),
    file(mboolean,k1_mboolean),
    []).

fof(dt_k2_closure1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & m3_pboole(D,A,B,C)
        & m2_pboole(E,A,B) )
     => m2_pboole(k2_closure1(A,B,C,D,E),A,C) ) ),
    file(closure1,k2_closure1),
    []).

fof(dt_k2_msualg_3,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => m3_pboole(k2_msualg_3(A,B),A,B,B) ) ),
    file(msualg_3,k2_msualg_3),
    []).

fof(dt_k3_closure1,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_pboole(B,A)
        & m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
        & m2_pboole(D,A,k7_mssubfam(A,B)) )
     => m2_pboole(k3_closure1(A,B,C,D),A,k7_mssubfam(A,B)) ) ),
    file(closure1,k3_closure1),
    []).

fof(dt_k7_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_mssubfam(k7_mssubfam(A,B),A,B)
        & v2_mssubfam(k7_mssubfam(A,B),A,B)
        & v3_mssubfam(k7_mssubfam(A,B),A,B)
        & v4_mssubfam(k7_mssubfam(A,B),A,B)
        & v5_mssubfam(k7_mssubfam(A,B),A,B)
        & v6_mssubfam(k7_mssubfam(A,B),A,B)
        & m4_pboole(k7_mssubfam(A,B),A,k1_mboolean(A,B)) ) ) ),
    file(mssubfam,k7_mssubfam),
    []).

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_m2_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m2_pboole(C,A,B)
         => m1_pboole(C,A) ) ) ),
    file(pboole,m2_pboole),
    []).

fof(dt_m3_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ! [D] : 
          ( m3_pboole(D,A,B,C)
         => m1_pboole(D,A) ) ) ),
    file(pboole,m3_pboole),
    []).

fof(dt_m4_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,B)
         => m1_pboole(C,A) ) ) ),
    file(pboole,m4_pboole),
    []).

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

fof(existence_m2_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ? [C] : m2_pboole(C,A,B) ) ),
    file(pboole,m2_pboole),
    []).

fof(existence_m3_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ? [D] : m3_pboole(D,A,B,C) ) ),
    file(pboole,m3_pboole),
    []).

fof(existence_m4_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ? [C] : m4_pboole(C,A,B) ) ),
    file(pboole,m4_pboole),
    []).

fof(rc1_funcop_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funcop_1(A) ) ),
    file(funcop_1,rc1_funcop_1),
    []).

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

fof(rc2_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ? [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C) ) ) ),
    file(mssubfam,rc2_mssubfam),
    []).

fof(rc3_pboole,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B) ) ),
    file(pboole,rc3_pboole),
    []).

fof(rc4_pboole,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_funcop_1(B) ) ),
    file(pboole,rc4_pboole),
    []).

fof(rc5_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc5_funct_1),
    []).

fof(rc5_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ? [C] : 
          ( m4_pboole(C,A,k1_mboolean(A,B))
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C)
          & v1_mssubfam(C,A,B)
          & v2_mssubfam(C,A,B)
          & v3_mssubfam(C,A,B)
          & v4_mssubfam(C,A,B)
          & v5_mssubfam(C,A,B)
          & v6_mssubfam(C,A,B) ) ) ),
    file(mssubfam,rc5_mssubfam),
    []).

fof(rc5_pboole,axiom,(
    ! [A,B] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ? [C] : 
          ( m4_pboole(C,A,B)
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C) ) ) ),
    file(pboole,rc5_pboole),
    []).

fof(redefinition_k2_closure1,axiom,(
    ! [A,B,C,D,E] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & m3_pboole(D,A,B,C)
        & m2_pboole(E,A,B) )
     => k2_closure1(A,B,C,D,E) = k15_pralg_1(D,E) ) ),
    file(closure1,k2_closure1),
    []).

fof(redefinition_k3_closure1,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_pboole(B,A)
        & m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
        & m2_pboole(D,A,k7_mssubfam(A,B)) )
     => k3_closure1(A,B,C,D) = k15_pralg_1(C,D) ) ),
    file(closure1,k3_closure1),
    []).

fof(redefinition_k7_mssubfam,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => k7_mssubfam(A,B) = k1_mboolean(A,B) ) ),
    file(mssubfam,k7_mssubfam),
    []).

fof(reflexivity_r2_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => r2_pboole(A,B,B) ) ),
    file(pboole,r2_pboole),
    []).

fof(t9_closure1,axiom,(
    ! [A,B] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ! [C] : 
          ( m2_pboole(C,A,B)
         => ! [D] : 
              ( m2_pboole(D,A,B)
             => ( r2_pboole(A,C,D)
               => r2_pboole(A,k2_closure1(A,B,B,k2_msualg_3(A,B),C),k2_closure1(A,B,B,k2_msualg_3(A,B),D)) ) ) ) ) ),
    file(closure1,t9_closure1),
    []).