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;

theorem Th60:
  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 dense & B is dense iff A is dense
proof
  let C, A be Subset of X, B be Subset of X0;
  assume
A1: C c= the carrier of X0;
  reconsider P = the carrier of X0 as Subset of X by TSEP_1:1;
  assume
A2: A c= C;
  assume
A3: A = B;
  thus C is dense & B is dense implies A is dense
  proof
    assume C is dense;
    then Cl C = [#]X;
    then
A4: [#]X c= Cl P by A1,PRE_TOPC:19;
    assume B is dense;
    then Cl B = [#]X0;
    then P = (Cl A) /\ [#]X0 by A3,PRE_TOPC:17;
    then Cl P c= Cl Cl A by PRE_TOPC:19,XBOOLE_1:17;
    then [#]X c= Cl A by A4,XBOOLE_1:1;
    then Cl A = [#]X;
    hence thesis;
  end;
  thus thesis by A2,A3,Th59,TOPS_1:44;
end;
