theorem
  D2 c= D1 & the TopStruct of X1 = the TopStruct of X2 implies (D1 is
  nowhere_dense implies D2 is nowhere_dense)
proof
  assume
A1: D2 c= D1;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  assume D1 is nowhere_dense;
  then Cl D1 is boundary;
  then Cl D2 is boundary by A1,A2,Th81,Th82;
  hence thesis;
end;
