reconsider e = x as Point of (Euclid n) by TOPREAL3:13;
A4: Sphere e,r is bounded by TOPREAL6:70;
Sphere e,r = Sphere x,r by Th15;
hence ( Sphere x,r is closed & Sphere x,r is Bounded ) by A4, JORDAN2C:def 2, TOPREAL6:69; :: thesis: verum