theorem LMBALL2:
  for p be Element of S, r be Real st 0 < r holds
    p in Ball(p,r) & p in cl_Ball(p,r)
  proof
    let p be Element of S, r be Real;
    assume
    A1: 0 < r;
    A2: ||. p - p .|| = ||. 0.S .|| by RLVECT_1:15
                     .= 0;
    hence p in Ball(p,r) by A1;
    thus p in cl_Ball(p,r) by A1,A2;
  end;
