Theorem:

The product of two even integers is also even.

Proof:

Let m and n be arbitrary integers.

Assume that both m and n are even.

By the definition of even integers, m = 2j holds for some integer j and n = 2k holds for some integer k.

Hence, the product of m and n is expressed as m ⋅ n = (2j) ⋅ (2k) = 2 ⋅ (2jk).

By the definition of even integers, the product m ⋅ n is even.

Therefore, for all even integers m and n, the product m ⋅ n is also even.