 
Summary: 1. The integers and the rational numbers.
1.1. The integers. Let
Z = {((m, n), (p, q)) (N × N) × (N × N) : m + q = p + n}.
It is a simple matter to verify that Z is an equivalence relation on N × N. Let
Z = (N × N)/Z
and call its members integers. Let
m  n = (m, n)/Z for (m, n) N × N.
One easily verifies that
N n (n  0)/Z
is univalent. In what follows we will not distinguish between a member of N and
its image under this mapping.
One verifies that there is a unique unary operation  on Z such that
(m  n) = n  m
and that there are unique binary operation + and on Z such that
(mn)+(pq) = (m+p)(n+q) and (mn)(pq) = (mp+nq)(mq+np)
for (m, n), (p, q) N × N; on the right hand side of these equations + and are the
operations on N.
One easily verifies that (Z, +, 0, ) is an Abelian group and that (Z, +, 0, ) is
an integral domain in which 1 is the neutral element with respect to .
1.2. The rational numbers. Let
