The conclusion of the implication is an existentially quantified statement.

To prove the conclusion using the proof by contradiction, we assume that we can put k+1 balls into k boxes such that no box contains 2 or more balls.

The statement to be proved is a universally quantified implication.

The previous statement contradicts the hypothesis that we can put k+1 balls into k boxes.

To prove the implication using the direct proof technique, we assume that we have put k+1 balls into k boxes.

Proof

Since there are k boxes and each box contains at most 1 ball, we can put at most k balls into these k boxes.

The assumption is not true and there must be a box that contains at least 2 balls.

To use the UG rule, we consider an arbitrary positive integer k.

QED

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