reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th22:
  Y1 is SubSpace of X1 & Y2 is SubSpace of X2 implies Y1 union Y2
  is SubSpace of X1 union X2
proof
  assume Y1 is SubSpace of X1 & Y2 is SubSpace of X2;
  then the carrier of Y1 c= the carrier of X1 & the carrier of Y2 c= the
  carrier of X2 by TSEP_1:4;
  then (the carrier of Y1) \/ (the carrier of Y2) c= (the carrier of X1) \/ (
  the carrier of X2) by XBOOLE_1:13;
  then the carrier of (Y1 union Y2) c= (the carrier of X1) \/ (the carrier of
  X2) by TSEP_1:def 2;
  then the carrier of (Y1 union Y2) c= the carrier of (X1 union X2) by
TSEP_1:def 2;
  hence thesis by TSEP_1:4;
end;
