reserve M for non empty MetrSpace,
  c,g1,g2 for Element of M;
reserve N for non empty MetrStruct,
  w for Element of N,
  G for Subset-Family of N,
  C for Subset of N;
reserve R for Reflexive non empty MetrStruct;
reserve T for Reflexive symmetric triangle non empty MetrStruct,
  t1 for Element of T,
  Y for Subset-Family of T,
  P for Subset of T;
reserve f for Function,
  n,m,p,n1,n2,k for Nat,
  r,s,L for Real,
  x,y for set;
reserve S1 for sequence of M,
  S2 for sequence of N;

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;
