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 Th17:
  for K be bounded finite-degree non void SimplicialComplex of TOP-REAL n st
      |.K.| c= [#]K
    holds diameter BCS(k,K) <= (degree K/(degree K+1))|^k * diameter K
  proof
  let K be bounded finite-degree non void SimplicialComplex of TOP-REAL n;
  set dK=degree K;
  set ddK=dK/(dK+1);
  defpred P[Nat] means
   diameter BCS($1,K)<=ddK|^$1*diameter K & BCS($1,K) is finite-degree & degree
BCS($1,K)<=dK;
  assume A1: |.K.|c=[#]K;
  A2: for k st P[k] holds P[k+1]
  proof
   let k;
   set T=TOP-REAL n;
   set B=BCS(k,K);
   set cM=center_of_mass T;
   A3: degree K>=-1 by SIMPLEX0:23;
   assume A4: P[k];
   then reconsider B=BCS(k,K) as bounded finite-degree non void
SimplicialComplex of TOP-REAL n;
   set dB=degree B;
   A5: degree B>=-1 by SIMPLEX0:23;
   A6: 0<=diameter K by Th7;
   A7: 0<=diameter B by Th7;
   A8: |.B.|=|.K.| & [#]B=[#]K by A1,SIMPLEX1:18,19;
   then A9: BCS B=subdivision(cM,B) by A1,SIMPLEX1:def 5;
   A10: BCS(k+1,K)=BCS B by A1,SIMPLEX1:20;
   then A11: diameter BCS(k+1,K)<=dB/(dB+1)*diameter B by A1,A8,Th16;
   not{} in dom cM by ORDERS_1:1;
   then dom cM is with_non-empty_elements by SETFAM_1:def 8;
   then A12: degree BCS(k+1,K)<=dB by A9,A10,SIMPLEX0:52;
   per cases;
   suppose dB=-1 or dB=0;
    then A13: dB/(dB+1)=0;
    per cases;
    suppose dK=0 or dK=-1;
     then dK/(dK+1)=0;
     then 0=(dK/(dK+1))|^(k+1) by NAT_1:11,NEWTON:11;
     hence thesis by A1,A4,A9,A11,A12,A13,SIMPLEX1:20,XXREAL_0:2;
    end;
    suppose A14: dK<>0 & dK<>-1;
     then dK>-1 by A3,XXREAL_0:1;
     then dK>=-1+1 by INT_1:7;
     then ddK>0 by A14,XREAL_1:139;
     then ddK|^(k+1)>0 by NEWTON:83;
     hence thesis by A1,A4,A6,A9,A11,A12,A13,SIMPLEX1:20,XXREAL_0:2;
    end;
   end;
   suppose dB<>-1 & dB<>0;
    then dB>-1 by A5,XXREAL_0:1;
    then A15: dB>=-1+1 by INT_1:7;
    A16: dB/(dB+1)=(dB+1)/(dB+1)-(1/(dB+1))
     .=1-(1/(dB+1)) by A15,XCMPLX_1:60;
    A17: ddK=(dK+1)/(dK+1)-(1/(dK+1))
     .=1-(1/(dK+1)) by A4,A15,XCMPLX_1:60;
    dB+1<=dK+1 by A4,XREAL_1:6;
    then 1/(dK+1)<=1/(dB+1) by A15,XREAL_1:85;
    then degree B/(degree B+1)<=dK/(dK+1) by A16,A17,XREAL_1:10;
    then A18: degree B/(degree B+1)*diameter B<=dK/(dK+1)*((dK/(dK+1))|^k*
diameter K) by A4,A7,A15,XREAL_1:66;
    dK/(dK+1)*((dK/(dK+1))|^k*diameter K)=dK/(dK+1)*(dK/(dK+1))|^k*(diameter K)
     .=(dK/(dK+1))|^(k+1)*(diameter K) by NEWTON:6;
    hence thesis by A1,A4,A9,A11,A12,A18,SIMPLEX1:20,XXREAL_0:2;
   end;
  end;
  ddK|^(0 qua Nat)=1 by NEWTON:4;
  then A19: P[0] by A1,SIMPLEX1:16;
  for k holds P[k] from NAT_1:sch 2(A19,A2);
  hence thesis;
 end;
