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

theorem Th30:
  X is extremally_disconnected iff for A being Subset of X st A is
  open holds Cl A = Int Cl A
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  open holds Cl A = Int Cl A
  proof
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
    A c= Cl A by PRE_TOPC:18;
    then
A2: A misses (Cl A)` by SUBSET_1:24;
    assume A is open;
    then (Cl A) misses Cl (Cl A)` by A1,A2,Th28;
    then Cl A c= (Cl (Cl A)`)` by SUBSET_1:23;
    then
A3: Cl A c= Int Cl A by TOPS_1:def 1;
    Int Cl A c= Cl A by TOPS_1:16;
    hence thesis by A3;
  end;
  assume
A4: for A being Subset of X st A is open holds Cl A = Int Cl A;
  now
    let A be Subset of X;
    assume A is open;
    then Cl A = Int Cl A by A4;
    hence Cl A is open;
  end;
  hence thesis;
end;
