Prove Theorem 1 as a corollary of Theorem 2.
Hint: add a source and sink and view the matching problem as a flow problem.
Theorem 1
A matching on a graph G is a maximum cardinality matching if and only if there is no augmenting path in G.
Theorem 2
A given feasible flow f on a graph maximizes the flow on (i, j) if and only if there is no augmenting path from j to i in the residual graph R(f).