We prove that if \(M\) is a maximal \(k\)-edge-colorable subgraph of a multigraph \(G\) and if \(F = \{v \in V(G) : d_M(v) \leq k-\mu(v)\}\), then \(d_F(v) \leq d_M(v)\) for all \(v \in F\). This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.