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 Th15:
  for a,ca be bounded Subset of Euclid n st ca = conv A & a = A holds
    diameter a = diameter ca
 proof
  let a,ca be bounded Subset of Euclid n;
  assume that
   A1: ca=conv A and
   A2: a=A;
  per cases;
  suppose a is empty;
   hence thesis by A1,A2;
  end;
  suppose A3: a is non empty;
   now let x,y be Point of Euclid n;
    assume x in ca;
    then A4: conv A c=cl_Ball(x,diameter a) by A1,A2,Th13;
    assume y in ca;
    hence dist(x,y)<=diameter a by A1,A4,METRIC_1:12;
   end;
   then A5: diameter ca<=diameter a by A1,A2,A3,TBSP_1:def 8;
   diameter a<=diameter ca by A1,A2,CONVEX1:41,TBSP_1:24;
   hence thesis by A5,XXREAL_0:1;
  end;
 end;
