reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;

theorem
  X1 is dense or X2 is dense implies X1 union X2 is dense SubSpace of X
proof
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
  reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  reconsider A = the carrier of X1 union X2 as Subset of X by TSEP_1:1;
  assume X1 is dense or X2 is dense;
  then A1 is dense or A2 is dense;
  then A1 \/ A2 is dense by TOPS_3:21;
  then A is dense by TSEP_1:def 2;
  hence thesis by Th9;
end;
