We present a short proof of a celebrated result of G\'acs and K\"orner giving sufficient and necessary condition on the joint distribution of two discrete random variables $X$ and $Y$ for the case when their mutual information matches the extractable (in the limit) common information. Our proof is based on the observation that the mere existence of certain random variables jointly distributed with $X$ and $Y$ can impose restriction on all random variables jointly distributed with $X$ and $Y$.

翻译：本文给出了Gács和Körner一个著名结果的简短证明，该结果给出了两个离散随机变量$X$和$Y$的联合分布在它们的互信息等于（极限上）可提取公共信息情况下的充分必要条件。我们的证明基于如下观察：某些与$X$和$Y$联合分布的随机变量的存在性本身，可以对所有与$X$和$Y$联合分布的随机变量施加限制。