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

theorem Th34:
  X is extremally_disconnected iff for A being Subset of X st A is
  condensed holds Int Cl A = Cl Int A
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  condensed holds Int Cl A = Cl Int A
  proof
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
    assume A is condensed;
    then
A2: Cl A = Cl Int A by Th9;
    Cl Int A is open by A1;
    hence thesis by A2,TOPS_1:23;
  end;
  assume
A3: for A being Subset of X st A is condensed holds Int Cl A = Cl Int A;
  now
    let A be Subset of X;
    assume
A4: A is condensed;
    then Int Cl A = Cl Int A by A3;
    hence A is closed & A is open by A4,Th8;
  end;
  hence thesis by Th32;
end;
