
theorem Th6:
  for h,s be non zero Nat
  st for q being Prime st q in support (prime_factorization s)
  holds not q,h are_coprime holds
  support (prime_factorization s) c= support (prime_factorization h)
  proof
    let h,s be non zero Nat;
    assume A1: for q being Prime st q in
    support (prime_factorization s) holds not q,h are_coprime;
    let x be object;
    assume A2: x in support prime_factorization s; then
    reconsider q=x as Prime by NEWTON:def 6;
    q divides h by Th5,A2,A1; then
    q in support (pfexp h) by ORDINAL1:def 13,NAT_3:37;
    hence thesis by NAT_3:def 9;
  end;
