reserve Y for TopStruct;

theorem Th25:
  (for P, Q being Subset of Y st P is closed & Q is closed holds P
  /\ Q is closed & P \/ Q is closed) implies for A, B being Subset of Y st A is
  closed & B is closed holds A is discrete & B is discrete implies A \/ B is
  discrete
proof
  assume
A1: for P,Q being Subset of Y st P is closed & Q is closed holds P /\ Q
  is closed & P \/ Q is closed;
  let A, B be Subset of Y;
  assume that
A2: A is closed and
A3: B is closed;
  assume that
A4: A is discrete and
A5: B is discrete;
  now
    let D be Subset of Y;
    D /\ A c= A by XBOOLE_1:17;
    then consider F1 being Subset of Y such that
A6: F1 is closed and
A7: A /\ F1 = D /\ A by A4;
    D /\ B c= B by XBOOLE_1:17;
    then consider F2 being Subset of Y such that
A8: F2 is closed and
A9: B /\ F2 = D /\ B by A5;
    assume D c= A \/ B;
    then
A10: D = D /\ (A \/ B) by XBOOLE_1:28;
    now
      take F = (A /\ F1) \/ (B /\ F2);
A11:  B /\ F2 is closed by A1,A3,A8;
      A /\ F1 is closed by A1,A2,A6;
      hence F is closed by A1,A11;
      thus (A \/ B) /\ F = D by A10,A7,A9,XBOOLE_1:23;
    end;
    hence ex F being Subset of Y st F is closed & (A \/ B) /\ F = D;
  end;
  hence thesis;
end;
