Once you have (P => Q) v (P => R), you reason by cases. If P => Q, then together with Q => R this gives P => R. Otherwise, you have P => R directly.
I've to show via a calculation that ((P => (Q v R)) ^ (Q => R)) => (P => R) is a tautology. Now, to show this, I've to show ((P => (Q v R)) ^ (Q => R)) |= (P =>R).
I've tried the following:
(P => (Q v R)) ^ (Q => R)
|= { by ^ v weakening }
P => (Q v R)
= { by distribution }
(P => Q) v (P => R)
But now I'm stuck, since ((P => Q) v (P => R)) is actually a weaker proposition than (P => R).
Obviously I'm doing something wrong. Should I apply monotonicity? Any help would be greatly appreciated!