assume
not OpenHypercube (e,r) is empty
; 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; verum