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 Th16:
  for K be bounded finite-degree non void SimplicialComplex of TOP-REAL n st
       |.K.| c= [#]K
    holds diameter BCS K <= degree K/(degree K+1) * diameter K
  proof
  set T=TOP-REAL n;
  let K be bounded finite-degree non void SimplicialComplex of T;
  set BK=BCS K;
  set cM=center_of_mass T;
  set dK=degree K;
  assume |.K.|c=[#]K;
  then A1: BCS K=subdivision(cM,K) by SIMPLEX1:def 5;
  for A be Subset of Euclid n st A is Simplex of BK holds diameter A<=dK/(dK+1)
*diameter K
  proof
   let A be Subset of Euclid n;
   reconsider ONE=1 as ExtReal;
   assume A2: A is Simplex of BK;
   then reconsider a=A as Simplex of BK;
   consider S be c=-linear finite simplex-like Subset-Family of K such that
    A3: A=cM.:S by A1,A2,SIMPLEX0:def 20;
   A4: A is bounded by A2,Th5;
   reconsider s=@S as c=-linear finite finite-membered Subset-Family of T;
   set Us=union s;
   set C=Complex_of{Us};
   union S is empty or union S in S by SIMPLEX0:9,ZFMISC_1:2;
   then A5: union S is simplex-like by TOPS_2:def 1;
   then A6: card union S<=degree K+ONE by SIMPLEX0:24;
   A7: diameter K>=0 by Th7;
   A8: not{} in dom cM by ORDERS_1:1;
   then A9: dom cM is with_non-empty_elements by SETFAM_1:def 8;
   A10: [#]T=[#]Euclid n by Lm1;
   then reconsider US=Us as Subset of Euclid n;
   A11: a in the topology of BK by PRE_TOPC:def 2;
   per cases;
   suppose K is empty-membered;
    then A12: dK=-1 by SIMPLEX0:22;
    then -1<=degree BK & degree BK<=-1 by A1,A9,SIMPLEX0:23,52;
    then degree BK=-1 by XXREAL_0:1;
    then BK is empty-membered by SIMPLEX0:22;
    then the topology of BK is empty-membered;
    then A13: a={} by A11,SETFAM_1:def 10;
    dK/(dK+1)=0 by A12;
    hence thesis by A13,TBSP_1:def 8;
   end;
   suppose K is non empty-membered;
    then degree K>=-1 & dK<>-1 by SIMPLEX0:22,23;
    then dK>-1 by XXREAL_0:1;
    then A14: dK>=-1+1 by INT_1:7;
    then A15: dK/(dK+1)*diameter K>=0 by A7;
    per cases;
    suppose a is empty;
     hence thesis by A15,TBSP_1:def 8;
    end;
    suppose A16: a is non empty;
     now US is bounded;
      then Us is bounded by JORDAN2C:11;
      then conv Us is bounded by Th14;
      then reconsider cUs=conv Us as bounded Subset of Euclid n by JORDAN2C:11;
      let x,y be Point of Euclid n;
      assume that
       A17: x in A and
       A18: y in A;
      reconsider X=x,Y=y as Element of T by A10;
      A19: |.BCS C.|=|.C.| & |.C.|=conv Us by SIMPLEX1:8,10;
      consider p be object such that
       A20: p in dom cM and
       A21: p in s and
       cM.p=x by A3,A17,FUNCT_1:def 6;
      reconsider p as set by TARSKI:1;
      p c=Us by A21,ZFMISC_1:74;
      then A22: Us<>{} by A8,A20;
      card Us<=dK+1 by A6,XXREAL_3:def 2;
      then A23: (dK+1)"<=(card Us)" by A22,XREAL_1:85;
      A24: diameter US<=diameter K by A5,Def4;
      A25: (card Us-1)/card Us=card Us/card Us-(1/card Us)
       .=1-(1/card Us) by A22,XCMPLX_1:60;
      A26: diameter cUs=diameter US by Th15;
      consider b1,b2 be VECTOR of T,r be Real such that
       A27: b1 in Vertices BCS C and
       A28: b2 in Vertices BCS C and
       A29: X-Y=r*(b1-b2) and
       A30: 0<=r and
       A31: r<=(card Us-1)/card Us by A3,A16,A17,A18,Th11;
      reconsider B1=b1,B2=b2 as Element of BCS C by A27,A28;
      B1 is vertex-like by A27,SIMPLEX0:def 4;
      then consider S1 be Subset of BCS C such that
       A32: S1 is simplex-like and
       A33: B1 in S1;
      A34: conv@S1 c=|.BCS C.| by A32,SIMPLEX1:5;
      B2 is vertex-like by A28,SIMPLEX0:def 4;
      then consider S2 be Subset of BCS C such that
       A35: S2 is simplex-like and
       A36: B2 in S2;
      @S2 c=conv@S2 by CONVEX1:41;
      then A37: B2 in conv@S2 by A36;
      @S1 c=conv@S1 by CONVEX1:41;
      then A38: B1 in conv@S1 by A33;
      reconsider bb1=b1,bb2=b2 as Point of Euclid n by A10;
      dK/(dK+1)=(dK+1)/(dK+1)-(1/(dK+1))
       .=1-(1/(dK+1)) by A14,XCMPLX_1:60;
      then (card Us-1)/card Us<=dK/(dK+1) by A23,A25,XREAL_1:10;
      then A39: dist(bb1,bb2)>=0 & r<=dK/(dK+1) by A31,XXREAL_0:2;
      conv@S2 c=|.BCS C.| by A35,SIMPLEX1:5;
      then dist(bb1,bb2)<=diameter US by A19,A26,A38,A37,A34,TBSP_1:def 8;
      then A40: dist(bb1,bb2)<=diameter K by A24,XXREAL_0:2;
      dist(x,y)=|.x-y.| by EUCLID:def 6
       .=|.X-Y.|
       .=|.r*(bb1-bb2).| by A29
       .=|.r.|*|.bb1-bb2.| by EUCLID:11
       .=r*|.bb1-bb2.| by A30,ABSVALUE:def 1
       .=r*dist(bb1,bb2) by EUCLID:def 6;
      hence dist(x,y)<=dK/(dK+1)*diameter K by A30,A40,A39,XREAL_1:66;
     end;
     hence thesis by A4,A16,TBSP_1:def 8;
    end;
   end;
  end;
  hence thesis by Def4;
 end;
