In Lang’s Algebra on pages 5-6 he mentions the property involving commutative monoids:
“Let $I, J$ be two sets, and $f:I\times J \rightarrow G$ a mapping into a commutative monoid which takes the value $e$ for almost all pairs $(i,j)$. Then
$$\prod_{i\in I}\prod_{j\in J}f(i,j)=\prod_{j\in J}\prod_{i\in I}f(i,j).”$$
By almost all he means all but a finite number of pairs $(i,j)$ and $e$ refers to the unit element. I’m well aware that all this entails of intuitively is arranging the terms rectangularly and then referring to them differently in the double product. I’m confused because I don’t know how specific of a proof this needs to be. I know I should use prior results in the book that state the terms in the product can be multiplied in any order, and that they can be parenthesized in any way while preserving the value. But I don’t know how to formulate it in a proof. Any help would be appreciated, especially a hint.