Numbers which Multiplied by 2 are the Reverse of when Added to 2

Theorem
... and so on:

Proof
We have that:
 * $\ds \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10^k + 7} + 2 = 4 \times 10^n + \sum_{k \mathop = 0}^{n - 1} 9 \times 10^k$

using the Basis Representation Theorem.

It remains to be demonstrated that:


 * $\ds 2 \times \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10^k + 7} = \sum_{k \mathop = 1}^n 9 \times 10^k + 4$

again using the Basis Representation Theorem.

Thus: