Modulo Addition is Closed

Note
On the face of it, there seems no need to prove this for both integers and reals.

From the above we see that the sum of two integers congruent to a given integer modulus is congruent to an integer to the same modulus.

However, it is important to note that the sum of two integers congruent to a given real modulus may not be congruent to an integer.

Take, as an example, $a = 2, b = 3, m = \pi$.

Then $\left({a + b}\right) \pmod \pi = 5 - \pi \notin \Z$.