Matthew wrote:

> As a note, Anindya remarks in comment #22 of Lecture 62, there may be issues with that lattice identity I have tried to use...

I don't see any problem with that. Suppose we have a poset \$$\mathcal{V}\$$ with all joins, and a doubly indexed family of elements in \$$\mathcal{V}\$$, say \$$\lbrace v_{a b} \rbrace_{a \in A, b \in B} \$$ where \$$A\$$ and \$$B\$$ are arbitrary sets. Then

$\bigvee_{a \in A} \bigvee_{b \in B} v_{ab} = \bigvee_{a \in A, b \in B} v_{ab} = \bigvee_{b \in B} \bigvee_{a \in A} v_{ab}$

It takes a bit of work to show this, but it seems so obvious I don't have the energy right now. All three are just different ways of thinking about the least upper bound of all the elements \$$v_{a b}\$$.

I guess in real analysis one should have seen this sort of fact when \$$\mathcal{V} = [-\infty,\infty] \$$ is the [extended real number line](https://en.wikipedia.org/wiki/Extended_real_number_line).

This fact implies a related fact for sums of arbitrary collections of numbers in \$$[0,\infty]\$$:

$\sum_{a \in A} \sum_{b \in B} v_{ab} = \sum_{a \in A, b \in B} v_{ab} = \sum_{b \in B} \sum_{a \in A} v_{ab}$

The reason is that these sums are defined as suprema of partial sums.