1+1 = 2

Theorem
Define $0$ as the only element in the set $P \setminus \map s P$, where:
 * $P$ is the Peano Structure
 * $\map s P$ is the image of the mapping $s$ defined in Peano structure
 * $\setminus$ denotes the set difference.

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

where:
 * $1 := \map s 0$
 * $2 := \map s 1 = \map s {\map s 0}$
 * $+$ denotes addition
 * $=$ denotes equality
 * $\map s n$ denotes the successor function as defined by Peano