theorem
  for S be finite Subset of M holds diameter(M,Complex_of{S}) = diameter S
 proof
  let S be finite Subset of M;
  set C=Complex_of{S};
  reconsider C as M bounded non void SimplicialComplex of M by Th6;
  reconsider s=S as Subset of C;
  A1: the topology of C=bool S by SIMPLEX0:4;
  now let W be Subset of M;
   assume W is Simplex of C;
   then W in bool S by A1,PRE_TOPC:def 2;
   hence diameter W<=diameter S by TBSP_1:24;
  end;
  then A2: diameter(M,C)<=diameter S by Def4;
  S c=S;
  then s is simplex-like by A1,PRE_TOPC:def 2;
  then diameter S<=diameter(M,C) by Def4;
  hence thesis by A2,XXREAL_0:1;
 end;
