  This paper presents two shortcuts to a classical proof of the
$\mathbf{M}_3$-$\mathbf{N}_5$ Theorem, which can be found in B. Davey and H.
Priestley [2] and S. Burris and H. Sankappanavar [1]. To be precise, the
shortcuts pertain a particular step of the proof that requires showing an
algebraic equality. In addition, we briefly discuss how to compare the lengths
of the three proofs (the original and our two proposed shortcuts). To do so, we
introduce two methods to compare the lengths of proofs based on algebraic
lattice expressions. We call them the proof count method and the proof poset
method. Both methods indicate that our proofs are shorter but the difference is
more pronounced in the former.
  Keywords: lattices, posets
