defpred S₁[ Nat] means ( BCS (V,K) is simplex-join-closed & BCS (V,K) is affinely-independent );
[#] 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 + 1),K) = BCS (BCS (n,K)) by A1, Th20;
hence S₁[n + 1] by A3; :: thesis: verum

end;

A4: S₁[ 0 ] by A1, Th16;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2);
hence BCS (n,K) is simplex-join-closed ; :: thesis: verum