reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;
reserve X for non empty TopSpace;

theorem
  X is extremally_disconnected iff for A, B being Subset of X st A is
closed & B is closed holds A \/ B = the carrier of X implies (Int A) \/ (Int B)
  = the carrier of X
proof
  thus X is extremally_disconnected implies for A, B being Subset of X st A is
closed & B is closed holds A \/ B = the carrier of X implies (Int A) \/ (Int B)
  = the carrier of X
  proof
    assume
A1: X is extremally_disconnected;
    let A, B be Subset of X;
    assume that
A2: A is closed and
A3: B is closed;
    assume A \/ B = the carrier of X;
    then (A \/ B)` = {}X by XBOOLE_1:37;
    then ( A`) /\ B` = {}X by XBOOLE_1:53;
    then ( A`) misses B`;
    then (Cl A`) misses (Cl B`) by A1,A2,A3,Th28;
    then (Cl A`) /\ (Cl B`) = {}X;
    then ((Cl A`) /\ (Cl B`))` = [#]X;
    then ( Cl A`)` \/ ( Cl B`)` = [#]X by XBOOLE_1:54;
    then ( Cl A`)` \/ (Int B) = [#]X by TOPS_1:def 1;
    hence thesis by TOPS_1:def 1;
  end;
  assume
A4: for A, B being Subset of X st A is closed & B is closed holds A \/
  B = the carrier of X implies (Int A) \/ (Int B) = the carrier of X;
  now
    let A, B be Subset of X;
    assume that
A5: A is open and
A6: B is open;
    assume A misses B;
    then A /\ B = {}X;
    then (A /\ B)` = [#]X;
    then A` \/ B` = [#]X by XBOOLE_1:54;
    then (Int A`) \/ (Int B`) = the carrier of X by A4,A5,A6;
    then ((Int A`) \/ (Int B`))` = {}X by XBOOLE_1:37;
    then (Int A`)` /\ (Int B`)` = {}X by XBOOLE_1:53;
    then (Cl A) /\ (Int B`)` = {}X by Th1;
    then (Cl A) misses (Int B`)`;
    hence (Cl A) misses (Cl B) by Th1;
  end;
  hence thesis by Th28;
end;
