theorem Th18:
  A is dense iff A` is boundary
proof
  thus A is dense implies A` is boundary
  proof
    assume A is dense;
    then ( A``) is dense;
    hence thesis by TOPS_1:def 4;
  end;
  assume A` is boundary;
  then ( A``) is dense by TOPS_1:def 4;
  hence thesis;
end;
