reconsider e = x as Point of (Euclid n) by TOPREAL3:13;
A1: n in NAT by ORDINAL1:def 13;
then ( Sphere (e,r) is bounded & Sphere (e,r) = Sphere (x,r) ) by Th15, TOPREAL6:70;
hence Sphere (x,r) is closed by A1, TOPREAL6:69; :: thesis: verum