% Mizar problem: t53_extreal2,extreal2,1172,26 
fof(t53_extreal2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ( A = k1_supinf_2
      <=> k4_extreal1(A) = k1_supinf_2 ) ) ),
    inference(mizar_bg_added,[status(thm)],[involutiveness_k3_supinf_2,reflexivity_r1_supinf_1,connectedness_r1_supinf_1,existence_m1_subset_1,dt_k1_supinf_2,dt_k3_supinf_1,dt_k3_supinf_2,dt_k4_extreal1,dt_m1_subset_1,t52_extreal2,d3_extreal1]),
    [file(extreal2,t53_extreal2)]).

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(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(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(t52_extreal2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k1_supinf_2
          & r1_supinf_1(k4_extreal1(A),k1_supinf_2) ) ) ),
    file(extreal2,t52_extreal2),
    []).

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),
    []).