reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;

theorem Th4:
  for M being non empty MetrSpace, z1, z2, z3 being Point of M st
  z1 <> z2 & z1 in cl_Ball(z3,r) & z2 in cl_Ball(z3,r) holds r > 0
proof
  let M be non empty MetrSpace, z1, z2, z3 be Point of M such that
A1: z1 <> z2 and
A2: z1 in cl_Ball(z3,r) and
A3: z2 in cl_Ball(z3,r);
  now
    assume r = 0;
    then
A4: cl_Ball(z3,r) = {z3} by TOPREAL6:56;
    then z1 = z3 by A2,TARSKI:def 1;
    hence contradiction by A1,A3,A4,TARSKI:def 1;
  end;
  hence thesis by A2,TOPREAL6:55;
end;
