reconsider e = x as Point of (Euclid n) by TOPREAL3:13;
A1: ( r is Real & Ball e,r is bounded ) by XREAL_0:def 1;
A2: n in NAT by ORDINAL1:def 13;
then Ball e,r = Ball x,r by Th13;
hence Ball x,r is open by A1, A2, GOBOARD6:6; :: thesis: verum