I don't understand the final sentence of the proof: why does \\( f(a \vee a') = f(a) \vee f(a') \\)?


So, \\( f(a \vee a') \\) is indeed the _least_ upper bound of \\( f(a) \\) and \\( f(a') \\). This implies that \\( f(a) \vee f(a') \\) actually exists, and that

\[ f(a \vee a') = f(a) \vee f(a'). \]