reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem
  R is dense & R c= S implies S is dense
proof
  assume that
A1: R is dense and
A2: R c= S;
  Cl R = [#] GX by A1;
  then [#] GX c= Cl S by A2,PRE_TOPC:19;
  then [#] GX = Cl S;
  hence thesis;
end;
