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
  closed holds Int A = Cl Int A
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  closed holds Int A = Cl Int A
  proof
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
    assume A is closed;
    then Cl A` = Int Cl A` by A1,Th30;
    then Int A = (Int ((Cl A`)``))` by TOPS_1:def 1;
    then Int A = Cl ( Cl A`)` by Th1;
    hence thesis;
  end;
  assume
A2: for A being Subset of X st A is closed holds Int A = Cl Int A;
  now
    let A be Subset of X;
    assume A is closed;
    then Int A = Cl Int A by A2;
    hence Int A is closed;
  end;
  hence thesis by Th27;
end;
