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

theorem Th1:
  q in OpenHypercube(p,r) & s in ].p.i - r, p.i + r.[
    implies q+*(i,s) in OpenHypercube(p,r)
proof
  assume that
A1: q in OpenHypercube(p,r)
  and
A2: s in ].p.i - r, p.i + r.[;
  consider e be Point of Euclid n such that
A3: p=e
  and
A4: OpenHypercube(p,r) = OpenHypercube(e,r) by Def1;
   set OH=OpenHypercube(e,r),I=Intervals(e,r), qs = q+*(i,s);
A5:  OH = product I by EUCLID_9:def 4;
   then
A6:  dom q = dom I by A4,A1,CARD_3:9;
A7:  dom I = dom e by EUCLID_9:def 3;
 A8: for x be object st x in dom I holds qs.x in I.x
    proof
      let x be object;
      assume
A9:     x in dom I;
      then
A10:    I.x = ].e.x-r,e.x+r.[ by A7,EUCLID_9:def 3;
A11:    q.x in I.x by A9,A1,A5,A4,CARD_3:9;
      i=x or i<>x;
      hence thesis by A6,A9,A3,FUNCT_7:31,32,A2,A10,A11;
    end;
  dom qs=dom q by FUNCT_7:30;
  hence thesis by A4,A8,CARD_3:9,A6,A5;
end;
