  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;

theorem Th8:
  n <= len R & r <= inf (rng R) implies
    OpenHypercube(p,r) c= ClosedHypercube(p,R)
proof
  set H=ClosedHypercube(p,R);
  assume that
A1: n <= len R
  and
A2: r <= inf rng R;
A3: Seg n c= Seg len R by A1,FINSEQ_1:5;
  set E=Euclid n,TE=TopSpaceMetr E,TR=TOP-REAL n;
  let x be object;
A4: Seg len R=dom R by FINSEQ_1:def 3;
  assume
A5:x in OpenHypercube(p,r);
  then reconsider q=x as Point of TR;
  consider e be Point of Euclid n such that
A6: p=e
  and
A7: OpenHypercube(p,r) = OpenHypercube(e,r) by Def1;
  set I=Intervals(e,r);
A8:x in product I by A5,A7,EUCLID_9:def 4;
  now
    let i;
A9: dom I = dom e by EUCLID_9:def 3;
    assume
A10:i in Seg n;
    then R.i in rng R by A3,A4,FUNCT_1:def 3;
    then inf (rng R) <= R.i by XXREAL_2:3;
    then
A11: r <= R.i by A2,XXREAL_0:2;
    then
A12: e.i +r <= e.i +R.i by XREAL_1:6;
A13: dom p = Seg len p by FINSEQ_1:def 3;
A14: e.i -r >= e.i - R.i by A11,XREAL_1:10;
A15: len p = n by CARD_1:def 7;
    then I.i = ].e.i-r,e.i+r.[ by A13, A6,A10,EUCLID_9:def 3;
    then
A16:q.i in ].e.i-r,e.i+r.[ by A9,CARD_3:9, A6,A10, A15,A13, A8;
    then e.i -r < q.i by XXREAL_1:4;
    then
A17: p.i - R.i < q.i by A14, A6,XXREAL_0:2;
    q.i < e.i +r by A16,XXREAL_1:4;
    then q.i < p.i + R.i by A12, A6,XXREAL_0:2;
    hence q.i in [.p.i-R.i,p.i+R.i.] by A17,XXREAL_1:1;
  end;
  hence thesis by Def2;
end;
