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

theorem Th37:
  for X being extremally_disconnected non empty TopSpace, X0
being non empty SubSpace of X, A being Subset of X st A = the carrier of X0 & A
  is dense holds X0 is extremally_disconnected
proof
  let X be extremally_disconnected non empty TopSpace, X0 be non empty
  SubSpace of X, A be Subset of X;
  assume
A1: A = the carrier of X0;
  assume
A2: A is dense;
  for D0, B0 being Subset of X0 st D0 is open & B0 is open holds D0 misses
  B0 implies (Cl D0) misses (Cl B0)
  proof
    let D0, B0 be Subset of X0;
    assume that
A3: D0 is open and
A4: B0 is open;
    consider D being Subset of X such that
A5: D is open and
A6: D /\ [#]X0 = D0 by A3,TOPS_2:24;
    consider B being Subset of X such that
A7: B is open and
A8: B /\ [#]X0 = B0 by A4,TOPS_2:24;
    assume
A9: D0 /\ B0 = {};
    D misses B
    proof
      assume D /\ B <> {};
      then (D /\ B) meets A by A2,A5,A7,TOPS_1:45;
      then (D /\ B) /\ A <> {};
      then D /\ (B /\ (A /\ A)) <> {} by XBOOLE_1:16;
      then D /\ (A /\ (B /\ A)) <> {} by XBOOLE_1:16;
      hence contradiction by A1,A6,A8,A9,XBOOLE_1:16;
    end;
    then (Cl D) misses (Cl B) by A5,A7,Th28;
    then (Cl D) /\ (Cl B) = {};
    then ((Cl D) /\ (Cl B)) /\ [#]X0 = {};
    then (Cl D) /\ ((Cl B) /\ ([#]X0 /\ [#]X0)) = {} by XBOOLE_1:16;
    then (Cl D) /\ ([#]X0 /\ ((Cl B) /\ [#]X0)) = {} by XBOOLE_1:16;
    then
A10: ((Cl D) /\ [#]X0) /\ ((Cl B) /\ [#]X0) = {} by XBOOLE_1:16;
A11: Cl B0 c= (Cl B) /\ [#]X0
    proof
      B0 c= B by A8,XBOOLE_1:17;
      then reconsider B1 = B0 as Subset of X by XBOOLE_1:1;
      Cl B1 c= Cl B by A8,PRE_TOPC:19,XBOOLE_1:17;
      then (Cl B1) /\ [#]X0 c= (Cl B) /\ [#]X0 by XBOOLE_1:26;
      hence thesis by PRE_TOPC:17;
    end;
    Cl D0 c= (Cl D) /\ [#]X0
    proof
      D0 c= D by A6,XBOOLE_1:17;
      then reconsider D1 = D0 as Subset of X by XBOOLE_1:1;
      Cl D1 c= Cl D by A6,PRE_TOPC:19,XBOOLE_1:17;
      then (Cl D1) /\ [#]X0 c= (Cl D) /\ [#]X0 by XBOOLE_1:26;
      hence thesis by PRE_TOPC:17;
    end;
    then (Cl D0) /\ (Cl B0) c= ((Cl D) /\ [#]X0) /\ ((Cl B) /\ [#]X0) by A11,
XBOOLE_1:27;
    then (Cl D0) /\ (Cl B0) = {} by A10;
    hence thesis;
  end;
  hence thesis by Th28;
end;
