Proof of *2.38 in Principia Mathematica (Russell/Whitehead)

Question

In Bertrand Russell and Alfred Whitehead's Principia Mathematica, they note that

"The proofs of *2.37.38 are exactly analogous to that of *2.36."

I have found a proof of *2.37, but would like to know a proof of *2.38, using the same format.

My proof of *2.37:

$Dem.$

1) $[Perm \frac{q,p}{p,q}] \vdash:q \vee p.\supset.p\vee q$

2) $[*2.06 \frac{q\vee p,p\vee q,p\vee r}{p,q,r}] \supset \vdash:.p\vee q.\supset. p\vee r:\supset:q\vee p.\supset. p\vee r$

3) $[Sum] \vdash:.q\supset r.\supset:p\vee q.\supset.p\vee r$

4) $\vdash.(2).(3).Syll.\supset\vdash.Prop$

Q.E.D.

Answer 1

Proof of 2.38:

1) $\vdash ((p \lor r) \supset (r \lor p)) \supset [((q \lor p) \supset (p \lor r)) \supset ((q \lor p) \supset (r \lor p))]$ --- (2.05 - Syll) with $(q \lor p)$ as $p$, $(p \lor r)$ as $q$ and $(r \lor p)$ as $r$

2) $\vdash (p \lor r) \supset (r \lor p)$ --- (1.5 - Perm)

3) $\vdash ((q \lor p) \supset (p \lor r)) \supset ((q \lor p) \supset (r \lor p))$ --- 1), 2), MP

4) $\vdash (q \supset r) \supset ((q \lor p) \supset (p \lor r))$ --- 2.37

5) $\vdash (q \supset r) \supset ((q \lor p) \supset (r \lor p))$ --- 4),3), Syll