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For every finite bipartite graph, the edge chromatic number equals its maximum degree. Repeatedly remove a matching covering maximum-degree vertices in a bipartite graph: each removal reduces the maximum degree by at least one. The removed matchings in a graph are colour classes of a proper edge colouring. The reverse bound follows from the distinct colours required at a vertex of maximum degree.

Let $A_{\Delta }=\{a\in A:d_G(a)=\Delta \}$. Apply Hall's marriage theorem to the induced subgraph on $A_{\Delta }\cup B$, which is a bipartite graph. For any subset $S\subseteq A_{\Delta }$, double counting the edges from $S$ to its graph neighbourhood givesThe equality uses the fact that every neighbour of $S$ lies in $B$, so all edges incident with $S$ have been counted. The final inequality uses the maximum degree bound on $B$. Since $\Delta >0$, division gives the Hall condition $|S|\le |N(S)|$.

Consequently there is a matching in a graph $M_1$ saturating $A_{\Delta }$. If $A_{\Delta }$ is empty, use the empty matching in a graph. This proves the required matching covering maximum-degree vertices in a bipartite graph on the specified side; other vertices need not be met.