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

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