General Distributivity Theorem/Warning

Worning about use of of General Distributivity Theorem
When using the General Distributivity Theorem, be careful not to conflate the indices.

The following is not a valid argument:

Proof
The second step uses the same index $i$ for two different purposes.

Thus elements of the form $\dfrac {a_i} {a_j}$ where $i \ne j$ are not represented in the sum.