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;

theorem
  for K be M bounded SimplicialComplexStr of X,
      KX be SubSimplicialComplex of K
    holds diameter(M,KX) <= diameter(M,K)
 proof
  let K be M bounded SimplicialComplexStr of X,KX be SubSimplicialComplex of K;
  A1: the topology of KX c=the topology of K by SIMPLEX0:def 13;
  per cases;
  suppose A2: the topology of KX meets bool[#]M;
   then the topology of K meets bool[#]M by A1,XBOOLE_1:63;
   then for A st A in the topology of KX holds diameter A<=diameter(M,K) by A1
,Def3;
   hence thesis by A2,Def3;
  end;
  suppose the topology of KX misses bool[#]M;
   then diameter(M,KX)=0 by Def3;
   hence thesis by Th7;
  end;
 end;
