## Theorem

The operation of addition on the set of integers $\Z$ is associative:

$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$

## Proof 1

From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

From Integers under Addition form Abelian Group, the integers under addition form a group, from which associativity follows from Group Axiom $\text G 1$: Associativity.

$\blacksquare$

## Proof 2

Let $a, b, c, d, e, f \in \N$ such that:

$x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$.

Then:

 $\ds x + \paren {y + z}$ $=$ $\ds \eqclass {a, b} {} + \paren {\eqclass {c, d} {} + \eqclass {e, f} {} }$ Definition of Integer $\ds$ $=$ $\ds \eqclass {a, b} {} + \eqclass {c + e, d + f} {}$ Definition of Integer Addition $\ds$ $=$ $\ds \eqclass {a + \paren {c + e}, b + \paren {d + f} } {}$ Definition of Integer Addition $\ds$ $=$ $\ds \eqclass {\paren {a + c} + e, \paren {b + d} + f} {}$ Natural Number Addition is Associative $\ds$ $=$ $\ds \eqclass {a + c, b + d} {} + \eqclass {e, f} {}$ Definition of Integer Addition $\ds$ $=$ $\ds \paren {\eqclass {a, b} {} + \eqclass {c, d} {} } + \eqclass {e, f} {}$ Definition of Integer Addition $\ds$ $=$ $\ds \paren {x + y} + z$ Definition of Integer

$\blacksquare$