Prove $(\Box(p\supset q)\land\Diamond(p\land r))\supset\Diamond(q\land r)$ in K $(\Box(p\supset q)\land\Diamond(p\land r))\supset\Diamond(q\land r)$
Here's what I have so far:

*

*$((p\supset q)\land(p\land r))\supset(q\land r)$, PC-valid WFF

*$\Box(((p\supset q)\land(p\land r))\supset(q\land r))$, N (1)

*$\Box((p\supset q)\land(p\land r))\supset\Box(q\land r)$, K (2)

*$\Box((p\supset q)\land(p\land r))\equiv(\Box(p\supset q)\land\Box(p\land r))$, K3 instance

*$(\Box(p\supset q)\land\Box(p\land r))\supset\Box(q\land r)$, Equiv (3),(4)

But now I'm stuck. This is problem 2.1c from Hughes and Cresswell. I think some instance of what they call K7 could be used:
$\Diamond(p\supset q)\equiv(\Box p\supset \Diamond q)$
Something like this:
$\Diamond(((p\supset q)\land(p\land r))\supset(q\land r))\equiv(\Box((p\supset q)\land(p\land r))\supset\Diamond(q\land r))$
But I don't know how to prove $\Diamond(((p\supset q)\land(p\land r))\supset(q\land r))$.
Thanks so much for any and all help.
 A: $\def\pra#1{\left(#1\right)}$
Hint: apply K$\Diamond$ and MP on step 2, instead of K, next try to find where you can apply K7, the K$\Diamond$ is
\begin{align}
&\square\pra{\varphi\supset\psi}\supset\pra{\Diamond\varphi\supset\Diamond\psi}\tag{K$\Diamond$}\\
1.&~~(\varphi\supset\psi)\supset(\lnot\psi\supset\lnot\varphi)\tag*{PC}\\
2.&~~\square(\varphi\supset\psi)\supset\square(\lnot\psi\supset\lnot\varphi)\tag*{N,K,MP(1)}\\
3.&~~\square(\lnot\psi\supset\lnot\varphi)\supset(\square\lnot\psi\supset\square\lnot\varphi)\tag*{K}\\
4.&~~\square(\varphi\supset\psi)\supset(\square\lnot\psi\supset\square\lnot\varphi)\tag*{PC (2,3)}\\
5.&~~\square(\varphi\supset\psi)\supset(\lnot\square\lnot\varphi\supset\lnot\square\lnot\psi)\tag*{PC(4)}
\end{align}
If you got this right, your final proof might end up with something like the following
\begin{align*}
&~~\square(p\supset q)\land\Diamond(p\land r)\supset\Diamond(q\land r)\\
1.&~~(p\supset q)\land(p\land r)\supset(q\land r)\tag*{PC}\\
2.&~~\square((p\supset q)\land(p\land r)\supset(q\land r))\tag*{N(1)}\\
3.&~~\Diamond((p\supset q)\land(p\land r))\supset\Diamond(q\land r)\tag*{K$\Diamond$,MP(2)}\\
4.&~~\square(p\supset q)\land\Diamond(p\land r)\supset\Diamond((p\supset q)\land(p\land r))\tag*{K7,PC,MP}\\
5.&~~\square(p\supset q)\land\Diamond(p\land r)\supset\Diamond(q\land r)\tag*{PC,MP(3,4)}
\end{align*}
