% Mizar problem: t102_extreal2,extreal2,2241,33 
fof(t102_extreal2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( k1_extreal2(A,k2_extreal2(A,B)) = A
            & k1_extreal2(k2_extreal2(A,B),A) = A
            & k1_extreal2(k2_extreal2(B,A),A) = A
            & k1_extreal2(A,k2_extreal2(B,A)) = A ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k1_extreal2,commutativity_k2_extreal2,connectedness_r1_supinf_1,d1_extreal2,d2_extreal2,dt_k1_extreal2,dt_k2_extreal2,dt_k3_supinf_1,dt_m1_subset_1,existence_m1_subset_1,idempotence_k1_extreal2,idempotence_k2_extreal2,reflexivity_r1_supinf_1,t95_extreal2]),
    [file(extreal2,t102_extreal2)]).

fof(commutativity_k1_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k1_extreal2(A,B) = k1_extreal2(B,A) ) ),
    file(extreal2,k1_extreal2),
    []).

fof(commutativity_k2_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal2(A,B) = k2_extreal2(B,A) ) ),
    file(extreal2,k2_extreal2),
    []).

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(d1_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(A,B)
             => k1_extreal2(A,B) = A )
            & ( ~ r1_supinf_1(A,B)
             => k1_extreal2(A,B) = B ) ) ) ) ),
    file(extreal2,d1_extreal2),
    []).

fof(d2_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(B,A)
             => k2_extreal2(A,B) = A )
            & ( ~ r1_supinf_1(B,A)
             => k2_extreal2(A,B) = B ) ) ) ) ),
    file(extreal2,d2_extreal2),
    []).

fof(dt_k1_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k1_extreal2(A,B),k3_supinf_1) ) ),
    file(extreal2,k1_extreal2),
    []).

fof(dt_k2_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k2_extreal2(A,B),k3_supinf_1) ) ),
    file(extreal2,k2_extreal2),
    []).

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

fof(idempotence_k1_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k1_extreal2(A,A) = A ) ),
    file(extreal2,k1_extreal2),
    []).

fof(idempotence_k2_extreal2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal2(A,A) = A ) ),
    file(extreal2,k2_extreal2),
    []).

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(t95_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal2(A,B) = A
            | k2_extreal2(A,B) = B ) ) ) ),
    file(extreal2,t95_extreal2),
    []).