If you still have the corresponding axioms, it should be pretty trivial to rebuild the idea of "combine these rules together to create significantly more complex rules", and then perhaps to relabel things in to "axioms" and "proofs". Leave a kid with a box of Legos and ey'll tend to build something, so the basic combination of "build by combination" seems pretty innate :)

If you've lost he explicit idea of axioms, but still have algebra, then you can get basic algebraic proofs, like 10X = 9X + 1X. If you play around from there, you should be able to come up with, and eventually prove, a few generalizations, and eventually you'll have a decent set of axioms. I'd expect you'd probably take a while to develop all of them.