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;

theorem Th22:
  n > 0 & |.Kv.| c= [#]Kv & degree Kv <= 0 implies
   the TopStruct of Kv = BCS(n,Kv)
 proof
  assume that
   A1: n>0 and
   A2: |.Kv.|c=[#]Kv and
   A3: degree Kv<=0;
  defpred P[Nat] means
   $1>0 implies the TopStruct of Kv=BCS($1,Kv) & degree BCS($1,Kv)<=0;
  A4: for n st P[n] holds P[n+1]
  proof
   not{} in dom(center_of_mass V) by ZFMISC_1:56;
   then A5: dom center_of_mass V is with_non-empty_elements;
   let n such that
    A6: P[n];
   assume n+1>0;
   per cases;
   suppose A7: n=0;
    A8: degree subdivision(center_of_mass V,Kv)<=degree Kv by A5,SIMPLEX0:52;
    BCS(n+1,Kv)=BCS Kv by A2,A7,Th17;
    hence thesis by A2,A3,A8,Def5,Th21;
   end;
   suppose A9: n>0;
    A10: |.Kv.|=|.BCS(n,Kv).| by Th10;
    [#]Kv=[#]BCS(n,Kv) by A6,A9;
    then BCS(n,Kv)=BCS BCS(n,Kv) by A2,A6,A9,A10,Th21;
    hence thesis by A2,A6,A9,Th20;
   end;
  end;
  A11: P[0 qua Nat];
  for n holds P[n] from NAT_1:sch 2(A11,A4);
  hence thesis by A1;
 end;
