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;

theorem Th40:
  A is nowhere_dense iff A` is everywhere_dense
proof
  thus A is nowhere_dense implies A` is everywhere_dense
  proof
    assume A is nowhere_dense;
    then Cl A is boundary;
    then Int Cl A = {}X;
    then Int (Int A`)` = {}X by TDLAT_3:1;
    then (Cl Int A`)` = {}X by TDLAT_3:3;
    then Cl Int A` = [#]X by Th2;
    hence thesis;
  end;
  assume A` is everywhere_dense;
  then Cl Int A` = [#]X;
  then (Cl Int A`)` = {}X by Th2;
  then Int (Int A`)` = {}X by TDLAT_3:3;
  then Int Cl A = {} by TDLAT_3:1;
  then Cl A is boundary;
  hence thesis;
end;
