### Zero sums

“Take some more tea,” the March Hare said to Alice, very earnestly. “I’ve had nothing yet,” Alice replied in an offended tone, “so I can’t take more.” “You mean you can’t take less,” said the Hatter: “it’s very easy to take

morethan nothing.” –from Alice’s Adventures in Wonderland

This is the trickiest challenge of our own adventures in reasoning about addition, and a proper capstone for this adventure.

[[Notes: this proof needs simplifications with T and F (or tautologies). Probably some earlier problem should introduce simplification. Also it uses some of the less intuitive properties of the conditional. And there should probably be a little discussion about irrelevant assumptions in an earlier page.]]