Nat Deduction Proof - Distributive property I need to prove (P v Q) ^ (P v R) |- P v (Q ^ R) using natural deduction and propositional logic.  I should be able to do it using only AND and OR rules, but I am stuck on how to assume Q and R.  This is what I have:
(P v Q) ^ (P v R)________premise
P v Q________________^e1___1
P v R________________^e2___1
...P__________________assumption
...P v (Q ^ R)___________vi1___4
I know this is right so far.  I just don't know how to introduce my assumptions for Q and R.  Do I assume ~P?  If so, which rule is applied, and what is the process, to conclude Q ^ R?  Thanks for any help you can give me

It depends on which rules you have for eliminating the disjunction.
If you indeed have the Disjunctive Syllogism, you can begin as Graham Kemp writes. But to complete, you will need Reductio ad Absurdum. From your question, however, I take it that you should not use that rule, and therefore not the Deductive Syllogism either.
The other standard option is that your disjunction elimination is a case analysis, which allows you to prove directly:
$P \vee Q, P \rightarrow R, Q \rightarrow R \vdash R$
To solve your problem, you need to apply this rule twice. Here is an outline. First, make two more assumptions, namely $Q$ and $R$. Second, introduce conjunction and disjunction to obtain $P \vee (Q \wedge R)$. Apply disjunction elimination for a first time (for example, on $P \vee R$, which will discharge $R$). Then apply it a second time, which will discharge the other assumption.
I can't write down the derivation since I don't know your variant of the calculus. That should be easy for you now, I hope.
