Help with Law of Excluded Middle I've got a problem with the law of excluded middle, and have a homework question surrounding it. I normally would never ask, and this is my first time, but I can't for the life of me find an example on my university's website with LEM in it. Here is the question: _Prove the validity of the sequents below (using LEM)_ $$ \vdash (p \to q) \lor (q \to r) $$

What rules of inference and/or axioms do you have to work with? If you have a natural deduction system, a possible proof [in Polish notation, Cxy means (x->y), Axy means (x V y), Nx means ~x, Kxy means (x ^ y)] might go)


1 ACpqNCpq form of the law of the excluded middle
2 | Cpq assumption
3 | ACpqCqr 2 alternation/disjunction-introduction
4 CCpqACpqCqr 2-3 Conditional Introduction
5 | NCpq assumption
6 || q assumption
7 ||| Nr assumption
8 |||| p assumption
9 |||| q 6, repetition
10 ||| Cpq 8-9 conditional introduction
11 ||| KCpqNCpq 10, 5 conjunction introduction
12 || r 7-11 negation elimination
13 | Cqr 6-12 conditional introduction
14 | ACpqCqr 13 alternation introduction
15 CNCpqACpqCqr 5-14 conditional introduction
16 ACpqCqr 1, 4, 15 alternation elimination.