LEMMA 3.15

Let K be a subfield of C, f an irreducible polynomial over K, and g, h polynomials over K. If g divides gh, then either f divides h or f divides h.

OK, so I have proven that f must divide over g or h - i.e., if f doesn't divide g, it must divide h - but it seems that f could still divide both, which is not what the text says.

f = ( x - 1 )

g = ( x - 1 )

^{2}( x - 2 )

h = ( x - 1 )

^{3}( x - 3 )

g h = ( x - 1 )

^{5}( x - 2 ) ( x - 3 )

Clearly, f divides ( g h ), g & h, so the LEMMA is wrong.

What am I missing here?