# “Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit confused.

Pardon the screenshot, but transcribing it would have been too tedious:

What is he getting at here? He first supposes that there is "only one independent simultaneous eigenbra" but then in the very next sentence he has a set of them. Does he actually mean one eigenbra per eigenvalue? That's what I think he means, but I thought I might put this up here for verification.

If you read the next paragraph, it seems that Dirac means that the eigenvalues are non-degenerate, i.e. for a set of simultaneous eigenvalues $\xi_1^\prime, \xi_2^\prime\dots$ there is exactly one corresponding eigenbra. There are no two distinct bras which are eigenbras of all the operators $\xi_1, \xi_2\dots$ and have the same eigenvalues for all of them.