In this paper we extend a lemma due to Yuan from several aspects. A new proof of Yuan’s lemma is given. A rank-one decomposition of positive semidefinite matrix is further developed. With the extended rank-one de- composition results, we generalize the Yuan’s lemma to general quadratic function systems, interval quadratic function systems and quadratic matrix function systems. Based on them, we offer several new proofs of S-lemma on quadratic functions systems, and establish a new S-lemma on quadratic matrix inequality functions systems and provide a simple proof of the strong duality of a class of quadratic matrix programming.