reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;

theorem Th21:
  A is bounded iff ex p be Point of Euclid n,r st A c= OpenHypercube(p,r)
 proof
  reconsider B=A as Subset of Euclid n by TOPREAL3:8;
  thus A is bounded implies ex p be Point of Euclid n,r be Real st A c=
OpenHypercube(p,r)
  proof
   assume A is bounded;
   then A1: B is bounded by JORDAN2C:11;
   per cases;
   suppose A2: A is empty;
    take p=the Point of Euclid n,r=the Real;
    thus thesis by A2;
   end;
   suppose A is non empty;
    then consider p be object such that
     A3: p in B by XBOOLE_0:def 1;
    reconsider p as Element of Euclid n by A3;
    consider r be Real such that
     0<r and
     A4: for x,y being Point of Euclid n st x in B & y in B holds dist(x,y)<=r
by A1;
    take p,r1=r+1;
    A5: B c=Ball(p,r1)
    proof
     let x be object;
     assume A6: x in B;
     then reconsider x as Element of Euclid n;
     dist(p,x)<r1 by A3,A4,A6,XREAL_1:39;
     hence thesis by METRIC_1:11;
    end;
    Ball(p,r1)c=OpenHypercube(p,r1) by EUCLID_9:22;
    hence A c=OpenHypercube(p,r1) by A5;
   end;
  end;
  given p be Point of Euclid n,r be Real such that
   A7: A c= OpenHypercube(p,r);
  per cases;
  suppose n=0; then
   OpenHypercube(p,r) = {{}} by EUCLID_9:12; then
   B = {} or B = {0} by A7,ZFMISC_1:33;
   hence thesis by JORDAN2C:11;
  end;
  suppose n<>0;
   then OpenHypercube(p,r)c=Ball(p,r*sqrt(n)) by EUCLID_9:18;
   then B is bounded by A7,TBSP_1:14,XBOOLE_1:1;
   hence thesis by JORDAN2C:11;
  end;
 end;
