Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 3/Proof 1

Proof
Let:

Also, let:

Hence:

Let:

as calculated above.

Thus:
 * $(1): \quad 2 A = A_1 + A_3$

Similarly:

Then:

Let:

as calculated above.

Thus:
 * $(2): \quad A_1 + A = A_2 + A_4$

Then:

Now we have that:

and so:

Finally: