I'm trying to derive:
Hall Theorem A bipartite graph G with partition (A,B) has a matching of A $\Leftrightarrow \forall S\subseteq A, |N(S)|\geq |S|$
From this:
Tutte Theorem A graph G has a 1-factor $\Leftrightarrow \forall S\subseteq V(G), q(G-S)\leq |S|$
where $q()$ denotes the number of odd connected components.
The idea of the proof is to suppose true the Tutte's condition for a bipartite graph G and by contradiction suppose that $\exists S\subseteq A,|N(S)|<|S|$
Now consider what happens if we remove from G the vertices of $N(S)$. All the vertex of $S$ become isolated vertex (odd component), and potentially we also have some other odd component, so we know that:
$|S|\leq q(G-N(S))\leq |N(S)|$
where we apply Tutte's theorem in the second inequality. Is this a valid proof?