reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve Ks for simplex-join-closed SimplicialComplex of V,
        As,Bs for Subset of Ks,
        Ka for non void affinely-independent SimplicialComplex of V,
        Kas for non void affinely-independent simplex-join-closed
                 SimplicialComplex of V,
        K for non void affinely-independent simplex-join-closed total
                 SimplicialComplex of V;

theorem Th30:
  |.Kv.| c= [#]Kv & (for n st n <= degree Kv ex S be Simplex of Kv st
     card S = n+1 & @S is affinely-independent)
  implies degree Kv = degree BCS Kv
 proof
  assume that
   A1: |.Kv.|c=[#]Kv and
   A2: for n st n<=degree Kv ex S be Simplex of Kv st card S=n+1 & @S is
affinely-independent;
  A3: dom center_of_mass V=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
  A4: for n st n<=degree Kv ex S be Subset of Kv st S is simplex-like & card S=
n+1 & BOOL S c=dom center_of_mass V & (center_of_mass V).:BOOL S
is Subset of Kv & (center_of_mass V)|BOOL S is one-to-one
  proof
   let n;
   assume n<=degree Kv;
   then consider S be Simplex of Kv such that
    A5: card S=n+1 and
    A6: @S is affinely-independent by A2;
   take S;
   thus S is simplex-like & card S=n+1 by A5;
   A7: the topology of Complex_of{@S}=bool S by SIMPLEX0:4;
   reconsider SS={@S} as affinely-independent Subset-Family of V by A6;
   A8: (center_of_mass V).:BOOL S c=conv@S
   proof
    let y be object;
    assume y in (center_of_mass V).:BOOL S;
    then consider x being object such that
     A9: x in dom(center_of_mass V) and
     A10: x in BOOL S & (center_of_mass V).x=y by FUNCT_1:def 6;
    reconsider x as non empty Subset of V by A9,ZFMISC_1:56;
    conv x c=conv@S & y in conv x by A10,RLAFFIN2:16,RLTOPSP1:20;
    hence thesis;
   end;
   bool@S c=(bool the carrier of V) by ZFMISC_1:67;
   hence BOOL S c=dom center_of_mass V by A3,XBOOLE_1:33;
   conv@S c=|.Kv.| by Th5;
   then conv@S c=[#]Kv by A1;
   hence (center_of_mass V).:BOOL S is Subset of Kv by A8,XBOOLE_1:1;
   (center_of_mass V)|bool S|BOOL S=(center_of_mass V)|BOOL S &
   Complex_of SS is SimplicialComplex of V by RELAT_1:74;
   hence thesis by A6,A7,FUNCT_1:52;
  end;
  not{} in dom center_of_mass V by ZFMISC_1:56;
  then A11: dom center_of_mass V is with_non-empty_elements;
  BCS Kv=subdivision(center_of_mass V,Kv) by A1,Def5;
  hence thesis by A4,A11,SIMPLEX0:53;
 end;
