Degree sum conditions on two disjoint cycles in graphs

Let G be a simple undirected graph on n vertices. A set of subgraphs of G is disjoint if no two of them have any common vertex in G. Suppose that n1,n2 are two integers with n1,n2≥3 and n=n1+n2. We prove that if d(x)+d(y)≥n+4 for any pair of vertices x,y of G with xy∉E(G), then G contains two disjoint cycles of length n1 and n2.