Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did we use?

1. Start with the given equations:
m + n = 2k
n + p = 2l

2. Solve these equations for \( m \) and \( p \):
m = 2k - n
p = 2l - n

3. Add \( m \) and \( p \):
m + p = (2k - n) + (2l - n)

4. Simplify the expression:
m + p = 2k + 2l - 2n
m + p = 2(k + l - n)

Since \( k + l - n \) is an integer, \( m + p \) is necessarily an even integer.

The type of proof used here is a direct proof. The answer is:

m + p = 2(k + l - n)