1. **State the problem:** Given the logical statements: 1. $P \to Q$ 2. $Q \to R$ Prove: $P \to R$ 2. **Formula used:** This is an example of the Hypothetical Syllogism rule in logic, which states: If $P \to Q$ and $Q \to R$, then $P \to R$. 3. **Explanation:** Hypothetical Syllogism allows us to chain implications. If the first statement implies the second, and the second implies the third, then the first implies the third. 4. **Intermediate work:** Let: - $A = P$ - $B = Q$ - $C = R$ Given: - $A \to B$ - $B \to C$ By Hypothetical Syllogism: $$A \to C$$ which means: $$P \to R$$ 5. **Final answer:** Therefore, from $P \to Q$ and $Q \to R$, we conclude $P \to R$ is valid by Hypothetical Syllogism.