  In deriving their characterization of the perfect matchings polytope,
Edmonds, Lov\'asz, and Pulleyblank introduced the so-called {\em Tight Cut
Lemma} as the most challenging aspect of their work. The Tight Cut Lemma in
fact claims {\em bricks} as the fundamental building blocks that constitute a
graph in studying the matching polytope and can be referred to as a key result
in this field. Even though the Tight Cut Lemma is a matching
\textup{(}$1$-matching\textup{)} theoretic statement that consists of purely
graph theoretic concepts, the known proofs either employ a linear programming
argument or are established upon results regarding a substantially wider notion
than matchings. This paper presents a new proof of the Tight Cut Lemma, which
attains both of the two reasonable features for the first time, namely, being
{\em purely graph theoretic} as well as {\em purely matching theory closed}.
Our proof uses, as the only preliminary result, the canonical decomposition
recently introduced by Kita. By further developing this canonical
decomposition, we acquire a new device of {\em towers} to analyze the structure
of bricks, and thus prove the Tight Cut Lemma. We believe that our new proof of
the Tight Cut Lemma provides a highly versatile example of how to handle
bricks.
