reserve
  x, y for object,
  i, n for Nat,
  r, s for Real,
  f1, f2 for n-element real-valued FinSequence;
reserve e, e1 for Point of Euclid n;

theorem Th10:
  for a, o being Point of Euclid n st a in Ball(o,r) holds
  for x being object holds |.(a-o).x.| < r & |.a.x-o.x.| < r
  proof
    let a, o be Point of Euclid n;
    reconsider a1 = a, o1 = o as Point of TOP-REAL n by EUCLID:67;
A1: Ball(o,r) = Ball(o1,r) by TOPREAL9:13;
    a-o = a1-o1;
    hence thesis by A1,Th9;
  end;
