reserve Y for TopStruct;

theorem Th24:
  (for P, Q being Subset of Y st P is open & Q is open holds P /\
Q is open & P \/ Q is open) implies for A, B being Subset of Y st A is open & B
  is open 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 open & Q is open holds P /\ Q is
  open & P \/ Q is open;
  let A, B be Subset of Y;
  assume that
A2: A is open and
A3: B is open;
  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 G1 being Subset of Y such that
A6: G1 is open and
A7: A /\ G1 = D /\ A by A4;
    D /\ B c= B by XBOOLE_1:17;
    then consider G2 being Subset of Y such that
A8: G2 is open and
A9: B /\ G2 = D /\ B by A5;
    assume D c= A \/ B;
    then
A10: D = D /\ (A \/ B) by XBOOLE_1:28;
    now
      take G = (A /\ G1) \/ (B /\ G2);
A11:  B /\ G2 is open by A1,A3,A8;
      A /\ G1 is open by A1,A2,A6;
      hence G is open by A1,A11;
      thus (A \/ B) /\ G = D by A10,A7,A9,XBOOLE_1:23;
    end;
    hence ex G being Subset of Y st G is open & (A \/ B) /\ G = D;
  end;
  hence thesis;
end;
