theorem Th12:
  for r being Real holds r<=0 implies Ball(t1,r) = {}
proof
  let r be Real;
  assume
A1: r<=0;
  set c = the Element of Ball(t1,r);
  assume
A2: Ball(t1,r) <> {};
  then reconsider c as Point of T by TARSKI:def 3;
  dist(t1,c)<r by A2,METRIC_1:11;
  hence contradiction by A1,METRIC_1:5;
end;
