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

theorem
  for p be Prime, k be Nat holds
    p divides Product primesFinS k iff primeindex p < k
proof
  let p be Prime, k be Nat;
A1:  p>1 by INT_2:def 4;
  thus p divides Product primesFinS k implies primeindex p < k
  proof
    assume p divides Product primesFinS k;
    then p |-count Product primesFinS k <> 0 by A1,NAT_3:27;
    hence thesis by Th28;
  end;
  assume primeindex p < k;
  then p |-count Product primesFinS k = 1 by Th28;
  hence thesis by A1,NAT_3:27;
end;
