Hi,

I've recently begun studying equational logic E, that outlined in Gries and Schneider's Logical Approach to Discrete math, and I'm working away proving each theorem in turn.

I've stumbled on theorem 3.45 (Distributivity of$\displaystyle \vee$ over $\displaystyle \wedge$): $\displaystyle p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$

and theorem 3.46 (Distributivity of$\displaystyle \wedge$ over $\displaystyle \vee$ ): $\displaystyle p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$

As usual, they can only be proven with theorems and axioms of a lower number, and truth tables aren't allowed. I keep starting off by going for axiom 3.35 (Golden rule): $\displaystyle p \wedge q \equiv p \equiv q \equiv p \vee q$, to transform (in the case of 3.45 obviously) $\displaystyle (p \vee q) \wedge (p \vee r)$ as my first step. Does anyone have any hints as to where to start? Am I headed in the right direction? How would anyone else prove these?

Thanks in advance!