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 SubSpace of X holds X0 is everywhere_dense iff ex X1
  being dense open strict SubSpace of X st X1 is SubSpace of X0
proof
  let X0 be SubSpace of X;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  thus X0 is everywhere_dense implies ex X1 being dense open strict SubSpace
  of X st X1 is SubSpace of X0
  proof
    assume X0 is everywhere_dense;
    then A is everywhere_dense;
    then consider C being Subset of X such that
A1: C c= A and
A2: C is open and
A3: C is dense by TOPS_3:41;
    C is non empty by A3,TOPS_3:17;
    then consider X1 being dense open strict non empty SubSpace of X such that
A4: C = the carrier of X1 by A2,A3,Th23;
    take X1;
    thus thesis by A1,A4,TSEP_1:4;
  end;
  given X1 being dense open strict SubSpace of X such that
A5: X1 is SubSpace of X0;
  reconsider C = the carrier of X1 as Subset of X by TSEP_1:1;
  now
    take C;
    thus C c= A & C is open & C is dense by A5,Def1,TSEP_1:4,16;
  end;
  then for A be Subset of X st A = the carrier of X0 holds A is
  everywhere_dense by TOPS_3:41;
  hence thesis;
end;
