1+1 = 2/Proof 1

Theorem
Define $0$ as the only element in the set $P \setminus s \left({P}\right)$, where:
 * $s \left({P}\right)$ is the image of the mapping $s$ defined in Peano structure
 * $\setminus$ denotes the set difference.

Then define $1$ as the successor of $0$, and $2$ as the successor of $1$.

The theorem to be proven is:
 * $1 + 1 = 2$

Proof
Define $1$ as $s \left({0}\right)$ and $2$ as $s \left({s \left({0}\right)}\right)$.

Therefore the statement to be proven becomes:


 * $s \left({0}\right) + s \left({0}\right) = s \left({s \left({0}\right)}\right)$

Thus: