Proof of Soundness

Assume the hypothesis (are partially correct tripples). Suppose input $x$ satisfies $\alpha$. Apply $P$ to it, then $Q$. Then after applying $P$ to $x$, any result satisfies $\beta$. So wehn $Q$ is applied to that, the result (if any) satisfies $\gamma$ (by the correctness of $\langle \beta \rangle \; Q \; \langle \gamma \rangle$). So by definition, $\langle \alpha \rangle \; P;Q \; \langle \gamma \rangle$ is partially correct.