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 dense non empty SubSpace of X, X2 being non empty
  SubSpace of X holds X1 is SubSpace of X2 implies X1 is dense SubSpace of X2
proof
  let X1 be 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 dense by Def1;
  then
A1: for A be Subset of X2 st A = the carrier of X1 holds A is dense by
TOPS_3:59;
  assume X1 is SubSpace of X2;
  hence thesis by A1,Def1;
end;
