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 Th13:
  for a be bounded Subset of Euclid n st a=A
  for p be Point of Euclid n st p in conv A holds
    conv A c= cl_Ball(p,diameter a)
  proof
  A1: the TopStruct of TOP-REAL n=TopSpaceMetr Euclid n by EUCLID:def 8;
  let a be bounded Subset of Euclid n
     such that
   A2: A=a;
  let x be Point of Euclid n such that
   A3: x in conv A;
  A4: A c=cl_Ball(x,diameter a)
  proof
   let p be object;
   assume A5: p in A;
   then reconsider p as Point of Euclid n by A2;
   reconsider pp=p as Point of TOP-REAL n by A1;
   A6: cl_Ball(p,diameter a)=cl_Ball(pp,diameter a) by TOPREAL9:14;
   A c=cl_Ball(p,diameter a)
   proof
    let y be object;
    assume A7: y in A;
    then reconsider q=y as Point of Euclid n by A2;
    dist(p,q)<=diameter a by A2,A5,A7,TBSP_1:def 8;
    hence thesis by METRIC_1:12;
   end;
   then conv A c=cl_Ball(pp,diameter a) by A6,CONVEX1:30;
   then dist(p,x)<=diameter a by A3,A6,METRIC_1:12;
   hence thesis by METRIC_1:12;
  end;
  reconsider xx=x as Point of TOP-REAL n by A1;
  cl_Ball(x,diameter a)=cl_Ball(xx,diameter a) by TOPREAL9:14;
  hence thesis by A4,CONVEX1:30;
 end;
