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 everywhere_dense or X2 is everywhere_dense implies X1 union X2
  is everywhere_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 everywhere_dense or X2 is everywhere_dense;
  then A1 is everywhere_dense or A2 is everywhere_dense;
  then A1 \/ A2 is everywhere_dense by TOPS_3:43;
  then A is everywhere_dense by TSEP_1:def 2;
  hence thesis by Th16;
end;
