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

theorem Th6:
  i in Seg n & q in ClosedHypercube(p,R) & r in [.p.i - R.i, p.i + R.i.]
    implies q+*(i,r) in ClosedHypercube(p,R)
proof
  set H=ClosedHypercube(p,R),pr=q+*(i,r);
  assume that
A1: i in Seg n
  and
A2: q in H
  and
A3: r in [.p.i - R.i, p.i + R.i.];
  for j be Nat st j in Seg n holds pr.j in [. p.j - R.j,p.j+R.j .]
  proof
    let j be Nat;
    assume
A4:   j in Seg n;
A5:   dom q = Seg len q by FINSEQ_1:def 3;
A6:   len q = n by CARD_1:def 7;
    per cases;
      suppose i <> j;
        then pr.j = q.j by FUNCT_7:32;
        hence thesis by Def2,A4,A2;
      end;
      suppose i =j;
        hence thesis by FUNCT_7:31,A1,A5,A6,A3;
      end;
  end;
  hence thesis by Def2;
end;
