Let $ \varphi $ be a $ \tau\_1 $-sentence, $ \psi $ be a $ \tau\_2 $-sentence. $ \tau=\tau\_1\cap\tau\_2 $. Suppose that $ \varphi\models\psi $. We need to show there is $ \tau $-sentence $ \theta $ such that $ \varphi\models\theta $ and $ \theta\models\psi $

Suppose $ \Gamma=\lbrace\sigma|\varphi\models\sigma,\sigma\ is\ a\ \tau\_1-sentence\rbrace $, We need to show $ \Gamma\models\psi $. For the contradiction, Suppose that $ \Gamma\nvDash\psi $. Let $ \cal M $ be a model such that $ \cal M\models$ $ \Gamma $$\cup\lbrace\neg\psi\rbrace $ . Let T=Th$\_\tau (\cal M) $=$ \lbrace\gamma|\cal M\models\gamma, $$\gamma$ is a $ \tau $- sentence$\rbrace $. Then 
T is a complete theory. So T$ \cup\lbrace\neg\psi\rbrace $ can be satisfied. If we can show that T$ \cup\lbrace\varphi\rbrace $ can be satisfied, then by Robinson Interpolation theorem, T$ \cup\lbrace\varphi\rbrace\cup \lbrace\neg\psi\rbrace $ can be satisfied. But it's contradict with $ \varphi\models\psi $. Then finished.

But how to show that T$ \cup\lbrace\varphi\rbrace $ can be satisfied?