You are asking two questions:
How can I prove that negative numbers are stored as pseudopositive?
Why are negative numbers stored as pseudopositive?
The answer to the first question is that no proof is needed, since one's complement is a way of representing signed numbers as unsigned numbers, and it specifies that negative numbers are stored in this specific way. There is nothing to prove here, just read the definition of one's complement. In other words, this is true by definition.
Regarding the second question, storing numbers in this way facilitates arithmetic. For example, to add two numbers, you just add them as unsigned numbers, with an extra little fiddling depending on the sign bits; in two's complement you don't even need the extra fiddling.