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 Th14:
  A is bounded iff conv A is bounded
  proof
  the TopStruct of TOP-REAL n=TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider cA=conv A,a=A as Subset of Euclid n;
  hereby assume A is bounded;
   then reconsider a as bounded Subset of Euclid n by JORDAN2C:11;
   set D=diameter a;
   A1: now let x,y be Point of Euclid n;
    assume x in cA;
    then A2: conv A c=cl_Ball(x,D) by Th13;
    assume y in cA;
    then dist(x,y)<=D by A2,METRIC_1:12;
    hence dist(x,y)<=D+1 by XREAL_1:39;
   end;
   D>=0 by TBSP_1:21;
   then cA is bounded by A1;
   hence conv A is bounded by JORDAN2C:11;
  end;
  assume conv A is bounded;
  then cA is bounded by JORDAN2C:11;
  then a is bounded by CONVEX1:41,TBSP_1:14;
  hence thesis by JORDAN2C:11;
 end;
