theorem
  A = cl_Ball(w,r) implies A is closed
proof
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider E=A as Subset of TopSpaceMetr Euclid n;
  assume A = cl_Ball(w,r);
  then E is closed by Th55;
  hence A is closed by A1,PRE_TOPC:31;
end;
