5.12 An interesting one.

Another which seems easy: . Let's see:

It's very easy to understand by anyone: it holds , but is false, so the truth is .

It can be done in several ways, but at some time you will have to use disjunction elimination to do something with the . We're going to prove that both and lead to the same place, which will be our target formula (since it's possible, let's go directly for ).

We open subdemonstration supposing that , and we must see that . It isn't too hard since we have on line 2; this helps contradicting anything we want. Since what we're searching is , we suppose and by reduction to the absurd we obtain , which is .

The other path, when we suppose true, leads us directly to .

In conclusion, both paths go to and by disjunction elimination we get the proof that is always certain.

Daniel Clemente Laboreo 2005-05-17