
theorem Th33:
  for V being RealUnitarySpace, v being Point of V, r,p being Real
  st r <= p holds Ball(v,r) c= Ball(v,p)
proof
  let V be RealUnitarySpace;
  let v be Point of V;
  let r,p be Real;
  assume
A1: r <= p;
  for u being Point of V st u in Ball(v,r) holds u in Ball(v,p)
  proof
    let u be Point of V;
    assume u in Ball(v,r);
    then dist(v,u) < r by BHSP_2:41;
    then dist(v,u) + r < r + p by A1,XREAL_1:8;
    then dist(v,u) < r + p - r by XREAL_1:20;
    hence thesis by BHSP_2:41;
  end;
  hence thesis;
end;
