  A graph $G$ with a perfect matching is called saturated if $G+e$ has more
perfect matchings than $G$ for any edge $e$ that is not in $G$. Lov\'asz gave a
characterization of the saturated graphs called the cathedral theorem, with
some applications to the enumeration problem of perfect matchings, and later
Szigeti gave another proof. In this paper, we give a new proof with our
preceding works which revealed canonical structures of general graphs with
perfect matchings. Here, the cathedral theorem is derived in quite a natural
way, providing more refined or generalized properties. Moreover, the new proof
shows that it can be proved without using the Gallai-Edmonds structure theorem.
