Theorem 4.5: Let m and n be any natural numbers.

• m < m.
• m < n and n < m, then m = n.
• If m < n and n < p then m < p.

Proof: a) m = m so

by Theorem 1.1, so m < m by Definition 4.6.

b) If m < n and n < m, then

and

by Definition 4.6, so m = n by Definition 1.2.

c) Suppose m < n and n < p. Then

and

by Definition 4.6, so

by Theorem 2.7, so m < p by Definition 4.6.