% Mizar problem: t45_extreal2,extreal2,1087,16 
fof(t45_extreal2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                    & r1_supinf_1(A,k1_supinf_2) )
                & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                    & r1_supinf_1(B,k1_supinf_2) ) )
              | ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                    & r1_supinf_1(k1_supinf_2,A) )
                & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                    & r1_supinf_1(k1_supinf_2,B) ) ) )
          <=> r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[dt_k1_extreal1,commutativity_k2_extreal1,reflexivity_r1_supinf_1,connectedness_r1_supinf_1,existence_m1_subset_1,redefinition_k2_extreal1,dt_k1_supinf_2,dt_k2_extreal1,dt_k3_supinf_1,dt_m1_subset_1,t44_extreal2]),
    [file(extreal2,t45_extreal2)]).

fof(dt_k1_extreal1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k1_extreal1(A,B),k3_supinf_1) ) ),
    file(extreal1,k1_extreal1),
    []).

fof(commutativity_k2_extreal1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal1(A,B) = k2_extreal1(B,A) ) ),
    file(extreal1,k2_extreal1),
    []).

fof(reflexivity_r1_supinf_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => r1_supinf_1(A,A) ) ),
    file(supinf_1,r1_supinf_1),
    []).

fof(connectedness_r1_supinf_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => ( r1_supinf_1(A,B)
        | r1_supinf_1(B,A) ) ) ),
    file(supinf_1,r1_supinf_1),
    []).

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

fof(redefinition_k2_extreal1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal1(A,B) = k1_extreal1(A,B) ) ),
    file(extreal1,k2_extreal1),
    []).

fof(dt_k1_supinf_2,axiom,(
    m1_subset_1(k1_supinf_2,k3_supinf_1) ),
    file(supinf_2,k1_supinf_2),
    []).

fof(dt_k2_extreal1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k2_extreal1(A,B),k3_supinf_1) ) ),
    file(extreal1,k2_extreal1),
    []).

fof(dt_k3_supinf_1,axiom,(
    $true ),
    file(supinf_1,k3_supinf_1),
    []).

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

fof(t44_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2)
                    & ~ r1_supinf_1(k1_supinf_2,B) )
                  | ( ~ r1_supinf_1(k1_supinf_2,A)
                    & ~ r1_supinf_1(B,k1_supinf_2) ) )
                & r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) )
            & ~ ( ~ r1_supinf_1(k1_supinf_2,k2_extreal1(A,B))
                & ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                    & ~ r1_supinf_1(k1_supinf_2,B) )
                & ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                    & ~ r1_supinf_1(B,k1_supinf_2) ) ) ) ) ) ),
    file(extreal2,t44_extreal2),
    []).