From Tutte's theorem to Hall's marriage theorem

Let $G$ be an $X,Y$-bigraph, and $|S|<o(G-S)$ for some $S\subseteq V(G)$. We aim to show that $|N(X')|<|X'|$ for some $X'\subseteq X$. This essentially proves Hall's marriage theorem using Tutte's theorem.

Let $X_0=S\cap X$ and $Y_0=S\cap Y$. Every odd component of $G-S$ has distinct sizes in $X$ and $Y$, so suppose that $G-S$ has $p$ odd components whose intersections with $X$ is larger than with $Y$, and $q$ odd components whose intersections with $X$ is smaller than with $Y$. Without loss of generality say $p>|Y_0|$. Let these $p$ odd components intersect with $X$ and $Y$ at $X_1,\dots,X_p$ and $Y_1,\dots,Y_p$, respectively. Define $$X'=X_1\cup\dots\cup X_p.$$ Then $$N(X')\subseteq Y_0\cup Y_1\cup\dots Y_p=Y'.$$ We have $$|X'|-|N(X')|\ge|X'|-|Y'|=|X_1|-|Y_1|+\dots+|X_p|-|Y_p|-|Y_0|\ge p-|Y_0|>0.$$