How to prove n+k'=(n+k)' where k' is the successor of k, and n is a natural number

(n+k)' is the successor of (n+k)

The successor to any natural number is defined to be that number obtained by adding 1 to the natural number in question.
Thus k +1 is the successor to the number k, and
(n+k)' = (n+k) + 1 is the successor.
Because of the associative property of addition,
(n+k) + 1 = n + (k+1)
Then substituting equals for equals gives
(n+k)' = n + k'
Then because the equality relationship is "reflexive", that is, if A = B, then B = A, so that
n + k' = (n + k)'.