reserve X for non empty TopSpace,
  D for Subset of X;

theorem Th5:
  for C being Subset of X modified_with_respect_to D st D c= C
  holds D is dense implies C is everywhere_dense
proof
  let C be Subset of X modified_with_respect_to D;
  assume
A1: D c= C;
  reconsider E = D as Subset of X modified_with_respect_to D by TMAP_1:93;
  assume
A2: D is dense;
  then
A3: E is open by Th4;
  E is dense by A2,Th4;
  hence thesis by A1,A3,TOPS_3:36,38;
end;
