Hints on Argument #39 and 41

#39: Theorem: Idempotence/v 39          Conclusion: A<->(AvA)

Once you have A->(AvA), you are half done.
Now, in order eventually apply <-> INTRO to it and another line, you need to derive:
(AvA)->A.  And to do this, it is unhelpful to assume A (your step 4).

#41  Theorem:Comutativity/v 41          Conclusion: (AvB)<->(BvA)

In 41, you don't really begin correctly.  Eventually, you'll want to apply <->INTRO (the
only rule that deals with '<->').  To do that, you need to derive:
(AvB)->(BvA), and then 
(BvA)->(AvB)
The strategies for obtaining both of these are straightforward, although the details are
a bit complicated to complete their individual derivations.

In both of these problems, (roughly, getting P->Q and Q->P, there are really two separate derivations, (roughly, getting P->Q and Q->P, then putting them together with <-> to get P<->Q),and one doesn't reach back to use a step in the first for the second, except for the <->INTRO step itself.

Both #39 and #41 follow the pattern of #40, but are harder--namely, in #39, one of the directions, say, Q->P, is harder and uses a different method than P->Q.
  In 41, BOTH directions are somewhat hard, but similar.