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;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;

theorem Th18:
  for K be bounded finite-degree non void SimplicialComplex of TOP-REAL n st
      |.K.| c= [#]K for r st r>0 ex k st diameter BCS(k,K) < r
 proof
  let K be bounded finite-degree non void SimplicialComplex of TOP-REAL n such
that
   A1: |.K.|c=[#]K;
  set dK=degree K;
  let r be Real such that
   A2: r>0;
  set ddK=dK/(dK+1);
  per cases;
  suppose dK=0 or dK=-1;
   then A3: ddK=0;
   diameter BCS K<=ddK*diameter K & BCS K=BCS(1,K) by A1,Th16,SIMPLEX1:17;
   hence thesis by A2,A3;
  end;
  suppose A4: diameter K=0;
   K=BCS(0,K) by A1,SIMPLEX1:16;
   hence thesis by A2,A4;
  end;
  suppose A5: dK<>0 & dK<>-1 & diameter K<>0;
   dK>=-1 by SIMPLEX0:23;
   then dK>-1 by A5,XXREAL_0:1;
   then A6: dK>=-1+1 by INT_1:7;
   then A7: ddK>0 by A5,XREAL_1:139;
   dK+1>dK by XREAL_1:29;
   then A8: ddK<1 by A6,XREAL_1:189;
   A9: diameter K>0 by A5,Th7;
   then r/diameter K>0 by A2,XREAL_1:139;
   then consider k be Nat such that
    A10: ddK to_power k<r/diameter K by A7,A8,TBSP_1:3;
   A11: r/diameter K*diameter K=r by A5,XCMPLX_1:87;
   A12: diameter BCS(k,K)<=ddK|^k*diameter K by A1,Th17;
   ddK to_power k=ddK|^k by POWER:41;
   then ddK|^k*diameter K<r by A9,A10,A11,XREAL_1:68;
   then diameter BCS(k,K)<r by A12,XXREAL_0:2;
   hence thesis;
  end;
 end;
