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How do I automatically label equations on the right? As below:

E=MC^2 Eq. (1)

I can make an autotitle for equations, and I have been able to get it to work on the left, but in my denseness I have not been able to do it as above.

You mean that instead of "Eq. (1) E=MC^2", you want to have it like this: "E=MC^2 Eq. (1)" ?

Hm, not sure how to do it. Maybe like this:

Well, I seem to have found a "solution," but it's a hack. The problem is that the equation number wants to be on a lower line because, if it's inserted to the right of the equation with autottile, it starts a new paragraph (unless there is a way to avoid that). To get the number to appear where it should be I had to define a paragraph style for the equation number AND a character style. I needed the latter in order to use the shift-from-base feature. Anyway, I can now get

E=MC^2 Eq. 1

but it's not pretty.