A holomorphic vector bundle $E\to M$ over a compact Kähler manifold $M$ with Kähler form $\omega$ is called ***stable*** if for any coherent analytic subsheaf $\mathcal F$ of lower rank of $E$ there holds $\mu(\mathcal F)<\mu(E)$ where $\mu(\mathcal F)=deg_\omega(\mathcal F)/rank(\mathcal F)$ and $deg_\omega(\mathcal F)=\int_MC_1(\mathcal F)\wedge *\omega$ where $C_1$ is the form representing the first Chern class.

My question is where can I find a proof (in case there is one) of the fact that such a stable holomorphic vector bundle is ***simple*** (which means that any holomorphic endomorphism is a constant times the identity). In the famous Uhlenbeck-Yau paper of 1986 it is said that "it is very likely" that it is true but they assume simplicity rather than giving a proof.