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

theorem Th12:
  OpenHypercube(p,r) c= ClosedHypercube(p,n|->r)
proof
  set TR= TOP-REAL n;
  per cases;
    suppose
A1:   n=0;
      then for i st i in Seg n holds
        (0.TR).i in [. p.i - (n|->r).i,p.i+(n|->r).i.];
      then
A2:   0.TR in ClosedHypercube(p,n|->r) by Def2;
      the carrier of TR = {0.TR} by JORDAN2C:105, A1,EUCLID:22;
      then ClosedHypercube(p,n|->r)=[#]TR by A2,ZFMISC_1:33;
      hence thesis;
    end;
    suppose n>0;
      then rng (n|->r) = {r} by FUNCOP_1:8;
      then
A3:     inf rng (n|->r) = r by XXREAL_2:13;
      len (n|->r) = n by CARD_1:def 7;
      hence thesis by Th8,A3;
    end;
end;
