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 Th24:
  X1 is SubSpace of X0 & X2 is SubSpace of X0 implies X1 union X2
  is SubSpace of X0
proof
  assume X1 is SubSpace of X0 & X2 is SubSpace of X0;
  then the carrier of X1 c= the carrier of X0 & the carrier of X2 c= the
  carrier of X0 by TSEP_1:4;
  then (the carrier of X1) \/ (the carrier of X2) c= the carrier of X0 by
XBOOLE_1:8;
  then the carrier of (X1 union X2) c= the carrier of X0 by TSEP_1:def 2;
  hence thesis by TSEP_1:4;
end;
