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 being Subset of X st A is
  condensed holds Int A = Cl A
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  condensed holds Int A = Cl A
  proof
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
    assume
A2: A is condensed;
    then A is closed by A1,Th32;
    then
A3: A = Cl A by PRE_TOPC:22;
    A is open by A1,A2,Th32;
    hence thesis by A3,TOPS_1:23;
  end;
  assume
A4: for A being Subset of X st A is condensed holds Int A = Cl A;
  now
    let A be Subset of X;
    assume A is condensed;
    then Int A = Cl A by A4;
    hence A is closed & A is open by Th5;
  end;
  hence thesis by Th32;
end;
