
theorem DivisorsMulti:
  for a,b being Nat holds
    PrimeDivisors (a * b) = PrimeDivisors a \/ PrimeDivisors b
  proof
    let a,b be Nat;
    thus PrimeDivisors (a * b) c= PrimeDivisors a \/ PrimeDivisors b
    proof
      let x be object;
      assume x in PrimeDivisors (a * b); then
      consider k being Prime such that
A1:   x = k & k divides a * b;
      k divides a or k divides b by A1,NAT_6:7; then
      x in PrimeDivisors a or x in PrimeDivisors b by A1;
      hence thesis by XBOOLE_0:def 3;
    end;
    let x be object;
    assume x in PrimeDivisors a \/ PrimeDivisors b; then
    per cases by XBOOLE_0:def 3;
    suppose x in PrimeDivisors a; then
      consider k being Prime such that
A1:   x = k & k divides a;
      k divides a * b by A1,INT_2:2;
      hence thesis by A1;
    end;
    suppose x in PrimeDivisors b; then
      consider k being Prime such that
A1:   x = k & k divides b;
      k divides a * b by A1,INT_2:2;
      hence thesis by A1;
    end;
  end;
