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

theorem
  r < s implies ClosedHypercube(p,n|->r) c= OpenHypercube(p,s)
proof
  assume
A1: r<s;
  let x be object such that
A2: x in ClosedHypercube(p,n|->r);
  reconsider q= x as Point of TOP-REAL n by A2;
    now
      let i such that
A3:     i in Seg n;
      (n|->r).i = r by A3,FINSEQ_2:57;
      then
A4:     q.i in [. p.i - r,p.i+r .] by A2,A3,Def2;
A5:   p.i + r < p.i + s by A1,XREAL_1:6;
      q.i <= p.i+r by A4,XXREAL_1:1;
      then
A6:     q.i < p.i+s by A5,XXREAL_0:2;
A7:     p.i - r > p.i - s by A1,XREAL_1:10;
      q.i >=p.i - r by A4,XXREAL_1:1;
      then q.i > p.i - s by A7,XXREAL_0:2;
      hence q.i in ]. p.i - s,p.i+s .[ by A6,XXREAL_1:4;
    end;
  hence thesis by Th3;
end;
