- (A1)
- .
- (A2)
- .

- By definition, .
- Suppose that is a natural number for which the sum is computable. Then we have , i.e. the sum is computable, i.e. .

- For , we have:

- Suppose that the property is true for some natural number . Then we have:

- For , the equality is obvious.
- Suppose that for some natural number . Then we have:

- For , the equality is obvious.
- Suppose that for some natural number . The we have:

- For any natural number , we have , by Lemma 1.11.
- Suppose that the equation holds for some natural number . Then we have:

- For , the property is obvious.
- Suppose that for some natural number and for any pair of natural numbers the given implication holds. Then we have:

- For , the property is obvious.
- Let ; then there exists a natural number such that . The we have:

If , then the predecessor of 0, and this does not exist. Therefore . We proved that if , then for any natural number , . This is the contrapositive expression of the desired result.