Using the direct proof technique, we assume x and y are both odd.

The statement to be proved is a universally quantified implication.

To use the UG rule, we consider two arbitrary integers x and y.

Using the definition/UI/MP/EI rules, we let x = 2m+1 and y=2n+1, where m and n are integers.

Proof

x+y = (2m+1) + (2n+1) = 2 \times (m+n+1).

Using the EG/definition/UI/MP rules, x+y is even.

QED