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

theorem
  for X1 being everywhere_dense SubSpace of X, X2 being SubSpace of X
  holds X1 is SubSpace of X2 implies X2 is everywhere_dense
proof
  let X1 be everywhere_dense SubSpace of X, X2 be SubSpace of X;
  reconsider C = the carrier of X1 as Subset of X by BORSUK_1:1;
  assume X1 is SubSpace of X2;
  then
A1: the carrier of X1 c= the carrier of X2 by TSEP_1:4;
  C is everywhere_dense by Def2;
  then for A be Subset of X st A = the carrier of X2 holds A is
  everywhere_dense by A1,TOPS_3:38;
  hence thesis;
end;
