reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve p for Prime;

theorem Th16:
  p,Product primesFinS k are_coprime implies primenumber(k) <= p
  proof
    set f = primesFinS k;
    set P = Product f;
    assume
A1: p,P are_coprime;
    assume primenumber(k) > p;
    then p in rng f by Th15;
    then p divides P by NAT_3:7;
    hence contradiction by A1,PYTHTRIP:def 2;
  end;
