reserve a,b,d,n,k,i,j,x,s for Nat;

theorem Th26:
  for k be Nat holds
     2|^k <= Product primesFinS k
proof
  defpred P[Nat] means 2|^$1 <= Product primesFinS $1;
A1: Product primesFinS 0 = 1 by RVSUM_1:94;
  2|^0 = 1 by NEWTON:4;
  then
A2: P[0] by A1;
A3: P[n] implies P[n+1]
  proof
    set n1=n+1;
    assume
A4:   P[n];
A5:   Product primesFinS n1 = Product (primesFinS n) * (primenumber n)
      by Th25;
    2 <= primenumber n by MOEBIUS2:8,21;
    then 2 * 2|^n <= Product primesFinS n1 by A5,A4,XREAL_1:66;
    hence thesis by NEWTON:6;
  end;
  P[n] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
