Remark

Prop. 1 applies with no restrictions to an arbitrary family $(x_i)$ of elements of $A$ (take $b = 1$), and also to any finite family $(x_i)$ of elements of $K$ (if $x_i = c_i b_i$ with $c_i \in A$ and $b_i \in A$, then take $b$ to be the product of the $b_i$).