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 subset-closed SimplicialComplexStr of X,
      i be dim-like number
    holds
  diameter(M,Skeleton_of(K,i)) <= diameter(M,K)
 proof
  set r = the Real;
  let K be M bounded subset-closed SimplicialComplexStr of X,
      i be dim-like number;
  set SK=Skeleton_of(K,i);
  A1: the topology of SK c=the topology of K by SIMPLEX0:def 13;
  per cases;
  suppose A2: the topology of SK meets bool[#]M;
   then A3: the topology of K meets bool[#]M by A1,XBOOLE_1:63;
   now let A;
    the_family_of K is subset-closed by MATROID0:def 3;
    then A4: the topology of K is subset-closed by MATROID0:def 1;
    assume A in the topology of SK;
    then consider w be set such that
     A5: A c=w and
     A6: w in the_subsets_with_limited_card(i+1,the topology of K) by
SIMPLEX0:2;
    reconsider w as Subset of K by A6;
    w in the topology of K by A6,SIMPLEX0:def 2;
    then A in the topology of K by A4,A5,CLASSES1:def 1;
    hence diameter A<=diameter(M,K) by A3,Def3;
   end;
   hence thesis by A2,Def3;
  end;
  suppose the topology of SK misses bool[#]M;
   then diameter(M,SK)=0 by Def3;
   hence thesis by Th7;
  end;
 end;
