Comment by wfo

Yet first order logic encompasses 99.9% of the mathematics anyone ever does, even professional mathematicians. The only people who consider anything but first order logic are people who have gone out of their way to make a point to study it just for its own sake. When the general subject of discussion is proof assistants and proofs in normal mathematics, I don't think it's necessary to constantly mention "only in first order logic". It's like discussing structural integrity of a building and constantly mentioning that all the calculations only hold on in our reality, but if we were living in an alternate dimension and the physical constants of the universe were different then the results wouldn't be applicable.