The question is entirely explained here, in that I wonder why every source seems to regard as obvious the claim that subprojections of finite projections are finite.
Here is the link.
To me, playing around with the tools and relations at hand doesn't work, and that's what would have to work before I'd make something a remark without further proof. So, I mean, I let $f$ be a subprojection of $e$ when $e$ is finite, and then if $g$ is a strict subprojection of $f$ then I need to show $g$ is not equivalent to $f$, so I notice $g$ is also a strict subprojection of $e$, hence is not equivalent to it, and then everything comes to a grinding halt. I must be missing something really easy.