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 Th5:
  for K be non void SimplicialComplexStr st
      K is M bounded & A is Simplex of K
    holds A is bounded
 proof
  let K be non void SimplicialComplexStr;
  assume K is M bounded;
  then A1: ex r be Real st for A st A in the topology of K
    holds A is bounded & diameter A<=r;
  assume A is Simplex of K;
  then A in the topology of K by PRE_TOPC:def 2;
  hence thesis by A1;
 end;
