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;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  X1 is nowhere_dense & X2 is boundary or X1 is boundary & X2 is
  nowhere_dense implies X1 union X2 is boundary
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;
  assume
  X1 is nowhere_dense & X2 is boundary or X1 is boundary & X2 is nowhere_dense;
  then A1 is nowhere_dense & A2 is boundary or A1 is boundary & A2 is
  nowhere_dense;
  then A1 \/ A2 is boundary by TOPS_3:30;
  then for A be Subset of X st A = the carrier of X1 union X2 holds A is
  boundary by TSEP_1:def 2;
  hence thesis;
end;
