1+1 = 2

Theorem
We will prove that $1 + 1 = 2$.

Proof
Define 1 as s(0) and 2 as s(s(0)). Therefore the statement to be proven becomes:
 * $s(0)+s(0)=s(s(0))$

Proof:

Proof 2
Defining $1$ as $s(0)$ and $2$ as $s(s(0))$ the statement to prove becomes:


 * $s(0) + s(0) = s(s(0))$

By the definition of addition:


 * $\forall m \forall n: m + s(n) = s(m + n)$

Letting $m = s(0)$ and $n = 0$:

By the definition of addition:


 * $\forall m: m + 0 = m$

Letting $m = s(0)$:


 * $s(0) + 0 = s(0)$

Taking the successor of both sides:

Applying Equality is Transitive to $(1)$ and $(2)$ we have:


 * $s(0) + s(0) = s(s(0))$

Hence the result.