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

theorem
  for X0 being everywhere_dense non empty SubSpace of X, A being Subset
  of X, B being Subset of X0 st A = B holds B is everywhere_dense iff A is
  everywhere_dense
proof
  let X0 be everywhere_dense non empty SubSpace of X, A be Subset of X, B be
  Subset of X0;
  assume
A1: A = B;
  reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
  C is everywhere_dense by Th16;
  hence thesis by A1,TOPS_3:64;
end;
