reserve r for Real;
reserve a, b for Real;
reserve T for non empty TopSpace;
reserve A for non empty SubSpace of T;
reserve P,Q for Subset of T,
  p for Point of T;

theorem Th4:
  T|P is SubSpace of T|(P \/ Q) & T|Q is SubSpace of T|(P \/ Q)
proof
  [#](T|P) = P & [#](T|(P \/ Q)) = P \/ Q by PRE_TOPC:def 5;
  hence T|P is SubSpace of T|(P \/ Q) by Th3,XBOOLE_1:7;
  [#](T|Q) = Q & [#](T|(P \/ Q)) = P \/ Q by PRE_TOPC:def 5;
  hence thesis by Th3,XBOOLE_1:7;
end;
