Hall's marriage theorem states that given a bipartite graph $G=(X+Y,E)$, if there is no $X$-saturating matching, there there exists $W\subseteq X$ such that $|W|>|N_G(W)|$.

Is the following generalized version true: if there is no matching that covers at least $|X|-k$ vertices of $X$, then there exists $W\subseteq X$ such that $|W|>|N_G(W)|+k$? I believe it is true by a similar proof as Hall's theorem, but it is not mentioned in the Wikipedia page. Is there a reference about this?