% Mizar problem: t57_extreal2,extreal2,1252,28 
fof(t57_extreal2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ( r1_supinf_1(k3_supinf_2(k4_extreal1(A)),A)
        & r1_supinf_1(A,k4_extreal1(A)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[connectedness_r1_supinf_1,d3_extreal1,dt_k1_supinf_2,dt_k3_supinf_1,dt_k3_supinf_2,dt_k4_extreal1,dt_m1_subset_1,existence_m1_subset_1,involutiveness_k3_supinf_2,reflexivity_r1_supinf_1,t16_supinf_2,t24_extreal1,t25_extreal1,t29_supinf_1,t51_extreal2]),
    [file(extreal2,t57_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(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_supinf_2,axiom,(
    m1_subset_1(k1_supinf_2,k3_supinf_1) ),
    file(supinf_2,k1_supinf_2),
    []).

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(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(t16_supinf_2,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(k3_supinf_2(B),k3_supinf_2(A)) ) ) ) ),
    file(supinf_2,t16_supinf_2),
    []).

fof(t24_extreal1,axiom,(
    k3_supinf_2(k1_supinf_2) = k1_supinf_2 ),
    file(extreal1,t24_extreal1),
    []).

fof(t25_extreal1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
            & r1_supinf_1(k1_supinf_2,k3_supinf_2(A)) )
        & ~ ( ~ r1_supinf_1(k1_supinf_2,k3_supinf_2(A))
            & r1_supinf_1(A,k1_supinf_2) )
        & ~ ( ~ r1_supinf_1(k1_supinf_2,A)
            & r1_supinf_1(k3_supinf_2(A),k1_supinf_2) )
        & ~ ( ~ r1_supinf_1(k3_supinf_2(A),k1_supinf_2)
            & r1_supinf_1(k1_supinf_2,A) ) ) ) ),
    file(extreal1,t25_extreal1),
    []).

fof(t29_supinf_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] : 
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] : 
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( r1_supinf_1(A,B)
                  & r1_supinf_1(B,C) )
               => r1_supinf_1(A,C) ) ) ) ) ),
    file(supinf_1,t29_supinf_1),
    []).

fof(t51_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => r1_supinf_1(k1_supinf_2,k4_extreal1(A)) ) ),
    file(extreal2,t51_extreal2),
    []).