Corollary 1

Let E be a monoid; if each of the elements $x_\alpha$ of an ordered sequence $(x_\alpha)_{\alpha \in A}$ of elements of E has an inverse $x'_\alpha$, the composition $\prod_{\alpha \in A} x_\alpha$ has inverse $\prod_{\alpha \in A'} x'_\alpha$, where A' is the totally ordered set derived from A by replacing the order on A by the opposite order.

This corollary follows from Proposition 4 by induction on the number of elements in A.

In particular, if x and x' are inverses, $\prod^n x$ and $\prod^n x'$ are inverses for every integer $n \geq 0$.