Conclusion Function
When setting up a proof, we put down the argument in the second section. But remember that we also put the conclusion in that proof, but we did not give it its own line. Why? We did this because the entire proof is meant to prove the conclusion. Simply, the conclusion cannot be part of the solution. We put the conclusion at the end of the last argument line to remind us of what we have to prove. We know we have successfully completed the proof when we can finally derive a conclusion that matches the desired conclusion in the second section.
In the example above, we see where the second section conclusion reminder is, and how the final derivation line has the same sentence (i.e. ~H) and we got that final derivation by going through the inference and replacements in order to get there.
It dawns on me that people may not understand why we do proofs in the first place. This is probably the most useful answer. A proof can be solved in any number of ways to get a conclusion (usually). But the simpler the proof (i.e. the shorter the proof length) the more coherent the argument.) If you have to do all sorts of inferences and what not’s to derive a conclusion from 40 something lines, the argument probably isn’t that coherent. There’s more to it, but that is just one advantage to these proofs.