An integer n is defined to be even when n=2k for some integer k.
An integer n is defined to be odd when n=2k +1 for some integer k.
Now we will look at a basic proof in abstract mathematics.
If x is even then x2 is even.
Notice that the statement above has an assumption and a conclusion. The assumption is that x is even while our conclusion is that x2 is even.
Now here is the formal proof.
Assume that x is even. [We wish to show that x2 is even]
By definition of even x=2k for some integer k.
Therefore, x2= (2k) 2= 4k2.
Moreover, x2 =4k2=2(2k2).
Since k is an integer then 2k2 is an integer. Hence, we conclude that x2 is even.