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;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem
  s <> t & s in closed_inside_of_circle(a,b,r) & t in
  closed_inside_of_circle(a,b,r) implies r > 0
proof
  reconsider x = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
  cl_Ball(x,r) = closed_inside_of_circle(a,b,r) by Th45;
  hence thesis by Th4;
end;
