I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" from Actes 20e Seminaire Lotharingien, p. 131–171. I'm fine with the paper up until p149 where MacDonald claims in result 3.9 the following:

$\omega_{q,t}E_{q,t}\omega_{q,t}^{-1}=E_{t^{-1},q^{-1}}$

Unfortunately MacDonald provides neither a proof nor a reference for this result. Instead he resorts to borrowing a margin from Fermat's book, claiming to have a proof while not actually giving the proof. Fermat told a better tale, though, claiming to have a "marvelous" proof whereas MacDonald admits to having only a "rather messy" proof.

Does anyone know where I can find MacDonald's "rather messy" proof, or better yet a more elegant proof of this result than the one MacDonald apparently had? Some of the results from this paper seem to also be in MacDonald's earlier book "Symmetric Functions and Hall Polynomials" but I can't find this one in that book either.

I'm trying to gain a rigorous understanding of this paper, but that is difficult for me to do if I need to accept MacDonald's apparent request that I simply gloss over this rather important intermediate step.