How to prove the existence of the identity element of an binary operator? In order to explain what I'm asking, let's consider the following binary operation: > The binary operation $*$ on $\mathbb{R}$ give by $x*y = x+y - 7$ for all $x,y$ $\in \mathbb{R}.$ In here it is pretty clear that the identity element exists and it is $7$, but in order to **prove** that the binary operation has the identity element $7$, first we have to prove the existence of an identity element than find what it is. So, how can we prove that the **existance** of the identity element ? Note: I actually asked a similar question before, but in that case the binary operation that I gave didn't have an identity element, so, as you can see from the answer, we directly proved with the method of contradiction.Therefore, instead of asking a new question, I'm editing my old question.

You guessed that the number $7$ acts as identity for the operation $*$. Then you checked that indeed $x*7=7*x=x$ for all $x$. Therewith **you have a full proof** that an identity element exists, and that $7$ is this special element.