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
everywhere_dense non empty SubSpace of X1 holds X2 is everywhere_dense SubSpace
  of X
proof
  let X1 be everywhere_dense non empty SubSpace of X, X2 be everywhere_dense
  non empty SubSpace of X1;
  reconsider C = the carrier of X1 as Subset of X by BORSUK_1:1;
  now
    let A be Subset of X;
    assume
A1: A = the carrier of X2;
    then reconsider B = A as Subset of X1 by BORSUK_1:1;
A2: C is everywhere_dense by Def2;
    B is everywhere_dense by A1,Def2;
    hence A is everywhere_dense by A2,TOPS_3:64;
  end;
  hence thesis by Def2,TSEP_1:7;
end;
