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 non empty SubSpace of X, X2 being non
  empty SubSpace of X holds X1 is SubSpace of X2 implies X1 is everywhere_dense
  SubSpace of X2
proof
  let X1 be everywhere_dense non empty SubSpace of X, X2 be non empty SubSpace
  of X;
  reconsider C = the carrier of X1 as Subset of X by BORSUK_1:1;
  C is everywhere_dense by Def2;
  then
A1: for A be Subset of X2 st A = the carrier of X1 holds A is
  everywhere_dense by TOPS_3:61;
  assume X1 is SubSpace of X2;
  hence thesis by A1,Def2;
end;
