**Puzzle 17**. We must show that if \$$S \subseteq T\$$ then \$$f(S) \subseteq f(T)\$$, which is equivalent to saying that \$$s \in f_\ast(S) \rightarrow s \in f_*(T)\$$. But if \$$s \in f_\ast(S)\$$ then \$$s = f(x)\$$ for some \$$x \in S\$$, which means \$$x \in T\$$, so \$$s \in f_\ast(T)\$$, as was to be shown.