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

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