What we've done here is this:
(a * key) + (b * key) = (c * key)
The rules of elementary algebra allow us to divide out the key on both sides because of a few useful symmetries that addition and multiplication have. Namely, these two equations are always the same number:
(a + b) * key = (a * key) + (b * key)
This is known as the distributive property. Normally, we talk about it applying to numbers being added and multiplied, but there are plenty of other mathematical structures and pairs of operations that do this, too. In the language of abstract algebra, we call any number system and pair of operations that distribute like this a "field", of which addition and multiplication over real[0] numbers is just one of.
A simple example of a field that isn't the normal number system you're used to is a 'finite field'. To visualize these, imagine a number circle instead of a line. We get a finite field by chopping off the number line at some prime[1] number that we decide is the highest in the loop. But this is still a field: addition and multiplication keep distributing.
It turns out cryptography loves using finite fields, so a lot of these identities hold in various cryptosystems. If I encrypt some data with RSA, which is just a pair of finite field exponents, multiplying that encrypted data will multiply the result when I decrypt it later on. In normal crypto, this is an attack we have to defend against, but in homomorphic crypto we want to deliberately design systems that allow manipulation of encrypted data like this in ways we approve of.
[0] Also complex numbers.
[1] Yes, it has to be prime and I'm unable to find a compact explanation as to why, I assume all the symmetries of algebra we're used to stop working if it's not.
