During the Ten Days of Repentance, we add a phrase in the 2nd to last blessing of Amidah:
וכתוב לחיים טובים כל בני בריתיך
"May all the children of your covenant be inscribed for good life."
Why is the term בני בריתיך used here as opposed to just simplifying it to כתבינו לחיים טובים ? (Yes, I'm aware that this would repeat what was said in the 1st blessing almost verbatim, but, we're already repeating the request to be inscribed for good life in all 4 of the insertions, anyway.)
I know what the term means, but, I am curious as to what additional idea is being conveyed, here, and, specifically, how that idea relates to this particular bracha. The term is not used in any of the other 3 places.