
theorem Th5:
  for h be non zero Nat, q being Prime
  st not q,h are_coprime holds
  q divides h
  proof
    let h be non zero Nat, q be Prime;
    set pq = prime_factorization q;
    set ph = prime_factorization h;
    A1: q=Product pq by NAT_3:61;
    A2: h=Product ph by NAT_3:61;
    assume not q,h are_coprime; then
    (support pq) /\ (support ph) <> {} by XBOOLE_0:def 7,INT_7:12,A1,A2;
    then (support pfexp q) /\ (support ph) <> {} by NAT_3:def 9;
    then (support pfexp q) /\ (support pfexp h) <> {} by NAT_3:def 9;
    then {q} /\ (support pfexp h) <> {} by NAT_3:43;
    then consider x be object such that
    A3: x in {q} /\ (support pfexp h) by XBOOLE_0:def 1;
    A4: x in {q} & x in (support pfexp h) by A3,XBOOLE_0:def 4;
    x=q by A4,TARSKI:def 1;
    hence q divides h by NAT_3:36,A4;
  end;
