theorem Th54:
  R is nowhere_dense implies R` is dense
proof
  assume R is nowhere_dense;
  then Cl R is boundary;
  then (Cl R)` is dense;
  then
A1: Cl((Cl R)`) = [#] GX;
  R c= Cl R by PRE_TOPC:18;
  then (Cl R)` c= R` by SUBSET_1:12;
  then [#] GX c= Cl(R`) by A1,PRE_TOPC:19;
  then [#] GX= Cl(R`);
  hence thesis;
end;
