% Mizar problem: t56_extreal2,extreal2,1198,30 
fof(t56_extreal2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => k4_extreal1(k2_extreal1(A,B)) = k2_extreal1(k4_extreal1(A),k4_extreal1(B)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k2_extreal1,connectedness_r1_supinf_1,d3_extreal1,dt_k1_extreal1,dt_k1_supinf_2,dt_k2_extreal1,dt_k3_supinf_1,dt_k3_supinf_2,dt_k4_extreal1,dt_m1_subset_1,existence_m1_subset_1,involutiveness_k3_supinf_2,redefinition_k2_extreal1,reflexivity_r1_supinf_1,t20_extreal1,t21_extreal1,t22_extreal1,t22_supinf_1,t26_extreal1]),
    [file(extreal2,t56_extreal2)]).

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(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(d3_extreal1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( r1_supinf_1(k1_supinf_2,A)
         => k4_extreal1(A) = A )
        & ( ~ r1_supinf_1(k1_supinf_2,A)
         => k4_extreal1(A) = k3_supinf_2(A) ) ) ) ),
    file(extreal1,d3_extreal1),
    []).

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(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_k3_supinf_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => m1_subset_1(k3_supinf_2(A),k3_supinf_1) ) ),
    file(supinf_2,k3_supinf_2),
    []).

fof(dt_k4_extreal1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => m1_subset_1(k4_extreal1(A),k3_supinf_1) ) ),
    file(extreal1,k4_extreal1),
    []).

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(involutiveness_k3_supinf_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => k3_supinf_2(k3_supinf_2(A)) = A ) ),
    file(supinf_2,k3_supinf_2),
    []).

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(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(t20_extreal1,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(B,k1_supinf_2) )
                | ( ~ r1_supinf_1(k1_supinf_2,A)
                  & ~ r1_supinf_1(k1_supinf_2,B) ) )
              & r1_supinf_1(k2_extreal1(A,B),k1_supinf_2) ) ) ) ),
    file(extreal1,t20_extreal1),
    []).

fof(t21_extreal1,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)) ) ) ) ),
    file(extreal1,t21_extreal1),
    []).

fof(t22_extreal1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal1(A,B) = k1_supinf_2
          <=> ( A = k1_supinf_2
              | B = k1_supinf_2 ) ) ) ) ),
    file(extreal1,t22_extreal1),
    []).

fof(t22_supinf_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(A,B)
              & r1_supinf_1(B,A) )
           => A = B ) ) ) ),
    file(supinf_1,t22_supinf_1),
    []).

fof(t26_extreal1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ( k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(A,k3_supinf_2(B))
            & k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(k3_supinf_2(A),B) ) ) ) ),
    file(extreal1,t26_extreal1),
    []).