reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;

theorem
  for C, A being Subset of X, B being Subset of X0 st C c= the carrier
of X0 & A c= C & A = B holds C is everywhere_dense & B is everywhere_dense iff
  A is everywhere_dense
proof
  let C, A be Subset of X, B be Subset of X0;
  assume
A1: C c= the carrier of X0;
  assume
A2: A c= C;
  assume
A3: A = B;
  thus C is everywhere_dense & B is everywhere_dense implies A is
  everywhere_dense
  proof
    Int C c= C by TOPS_1:16;
    then reconsider D = Int C as Subset of X0 by A1,XBOOLE_1:1;
A4: D /\ B c= Int C by XBOOLE_1:17;
    then reconsider E = D /\ B as Subset of X by XBOOLE_1:1;
    assume
A5: C is everywhere_dense;
    then Int C is everywhere_dense;
    then
A6: D is everywhere_dense by Th61;
    assume B is everywhere_dense;
    then
A7: D /\ B is everywhere_dense by A6,Th44;
    Int C is dense by A5;
    then E is everywhere_dense by A7,A4,Th62;
    hence thesis by A3,Th38,XBOOLE_1:17;
  end;
  thus thesis by A2,A3,Th38,Th61;
end;
