defpred S₁[ Nat] means BCS (n,K) is finite-vertices ;
[#] K = [#] V by SIMPLEX0:def 10;
then A1: |.K.| c= [#] K ;
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
[#] (BCS (n,K)) = [#] V by SIMPLEX0:def 10;
then A4: |.(BCS (n,K)).| c= [#] (BCS (n,K)) ;
BCS ((n + 1),K) = BCS (BCS (n,K)) by A1, Th20
.= subdivision ((center_of_mass V),(BCS (n,K))) by A4, Def5 ;
hence S₁[n + 1] by A3; :: thesis: verum

end;

A5: S₁[ 0 ] by A1, Th16;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2);
hence BCS (n,K) is finite-vertices ; :: thesis: verum