A proof of Slight Deficiency

Recently, I came across a Wolfram Math World
article, which was about slightly deficient numbers.
These were numbers whose factors added up to one
less than the number itself. For example:-

32's factors --> 1, 2, 4, 8, 16 --> 1 + 2 + 4 + 8 + 16 = 31 = 32 - 1

So, 32 is slightly deficient.

The Wolfram Math World article stated that it was
an unproven conjecture that all powers of 2 were
slightly deficient. I decided to attempt a proof.

I believe that I have succeeded, because I could not
find any errors in the proof. However, if I have made
a mistake, then please leave a comment saying so.

If the image, which was originally in pdf form, is too
hard to read, then I will change it. Please leave a comment
saying so, if this is the case. Thank you.

The proof is below.