I think this could use a bit more elaboration, since if x-y+y-z < -(|x-y|+|y-z|), then ||x-y|+|y-z|| >= |x-y+y-z| wouldnt be true. This is impossible though since q >= -|q|

I see, thanks! :3

I'm confused about this step in the final condition's proof:

|🍎(x) -🍌(x)| +|🍌(x) - 🍇(x)| >=|🍎(x) -🍌(x) +🍌(x) - 🍇(x)| = |🍎(x) - 🍇(x)| since |q| >= q forall q

I can see how it's true by proving that |p| + |q| >= |p + q|, but that's not stated anywhere and I can't figure out how |q| >= q forall q is relevant.

Also, thanks a lot for making/showing a proof :D

i saw this before going to bed and i dreampt mojang actually added this and i was watching hbg highlights videos in the new update