assume not OpenHypercube e,r is empty ; :: thesis: contradiction
then consider x being set such that
A1: x in OpenHypercube e,r by XBOOLE_0:def 1;
reconsider x = x as Point of (TopSpaceMetr (Euclid n)) by A1;
reconsider e1 = e, x = x as real-valued Function ;
set f = Intervals e,r;
A2: dom (Intervals e,r) = dom e by Def3;
A3: dom e = Seg n by EUCLID:3;
consider N being set such that
A4: N in Seg n by XBOOLE_0:def 1;
(Intervals e,r) . N = ].((e1 . N) - r),((e1 . N) + r).[ by A4, A3, Def3;
hence contradiction by A1, A2, A3, A4, CARD_3:18; :: thesis: verum