reconsider e = x as Point of (Euclid n) by TOPREAL3:8;
A1: ( r is Real & Ball (e,r) is bounded ) by XREAL_0:def 1;
n in NAT by ORDINAL1:def 12;
then Ball (e,r) = Ball (x,r) by Th13;
hence Ball (x,r) is open by A1, GOBOARD6:3; :: thesis: verum