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

theorem Th4:
  (ex i st i in Seg n /\ dom R & R.i < 0) implies ClosedHypercube(p,R) is empty
proof
  given i such that
A1:   i in Seg n /\ dom R
  and
A2:   R.i < 0;
  assume ClosedHypercube(p,R) is non empty;
  then consider x be object such that
A3: x in ClosedHypercube(p,R);
  reconsider x as Point of TOP-REAL n by A3;
  i in Seg n by A1,XBOOLE_0:def 4;
  then
A4: x.i in [. p.i - R.i,p.i+R.i .] by Def2,A3;
  then
A5: x.i <= p.i+R.i by XXREAL_1:1;
  p.i - R.i <= x.i by A4,XXREAL_1:1;
  then p.i +- R.i <= p.i +R.i by A5,XXREAL_0:2;
  hence contradiction by A2,XREAL_1:8;
end;
