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 Th6:
  K is finite-vertices implies K is M bounded
  proof
  set V=Vertices K;
  assume K is finite-vertices;
  then V is finite;
  then reconsider VM=V/\[#]M as finite Subset of M;
  take diameter VM;
  let A;
  assume A1: A in the topology of K;
  then reconsider S=A as Subset of K;
  S is simplex-like by A1,PRE_TOPC:def 2;
  then A c=V by SIMPLEX0:17;
  then A c=VM by XBOOLE_1:19;
  hence thesis by TBSP_1:24;
 end;
