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