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 Th7:
  K is M bounded implies 0 <= diameter(M,K)
 proof
  assume A1: K is M bounded;
  per cases;
  suppose A2: the topology of K meets bool[#]M;
   then consider S be object such that
    A3: S in the topology of K and
    A4: S in bool[#]M by XBOOLE_0:3;
   reconsider S as Subset of M by A4;
   ex r be Real st for A st A in the topology of K holds A is bounded &
diameter A<=r by A1;
   then diameter S>=0 by A3,TBSP_1:21;
   hence thesis by A1,A2,A3,Def3;
  end;
  suppose the topology of K misses bool[#]M;
   hence thesis by A1,Def3;
  end;
 end;
