Axioms are a set of statements upon which other statements can be formed.
Lemmas are easy but irrelevant.
Corollaries are quick but irrelevant consequences
Propositions are interesting
Theorems are important and difficult
https://www.quora.com/What-is-the-difference-between-a-lemma-theorem-corollary-and-proposition
http://blogs.scienceforums.net/ajb/2013/01/12/lemma-theorem-proposition-or-corollary/
https://math.stackexchange.com/questions/463362/whats-the-difference-between-theorem-lemma-and-corollary
A proof is about consistency of statements with respect to a set of axioms.
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To show the connection can be made, pick the simplest PDG inference rules.
* "add X to both sides"
* "multiply both sides by"
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