reserve X, Y for RealNormSpace;

theorem Th14:
  for x be Point of X, r, s be Real st r <= s holds Ball(x,
  r) c= Ball(x,s)
proof
  let x be Point of X, r, s be Real;
  assume
A1: r <=s;
  for u being Point of X st u in Ball(x,r) holds u in Ball(x,s)
  proof
    let u be Point of X;
    assume u in Ball(x,r);
    then ex uu1 be Point of X st u=uu1 & ||.x-uu1.|| < r;
    then ||.x-u.|| + r < r + s by A1,XREAL_1:8;
    then ||.x-u.|| < r + s - r by XREAL_1:20;
    hence thesis;
  end;
  hence thesis;
end;
