assume not OpenHypercube (e,r) is empty ; :: thesis: contradiction
then consider x being object such that
A1: x in OpenHypercube (e,r) ;
reconsider e1 = e as real-valued Function ;
set f = Intervals (e,r);
A2: dom (Intervals (e,r)) = dom e by Def3;
A3: dom e = Seg n by FINSEQ_1:89;
consider N being object 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_3:9; :: thesis: verum