@Charles Clingen

(You can quote a piece of text by putting > and a space before each line. It seems to work even for TeX.)

> 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

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

I believe that step is using the definition of join, no fancy tricks.

\\(f(a) \vee f(a')\\)

\\(= \textrm{The join is defined as the least upper bound of the two objects.}\\)

\\(= f(a \vee a') \textrm{, which was shown to be the least upper bound.}\\)