Uniqueness of Identity and Multiplication by 0

Theorem

Let $a,b,u,z\in \mathbb{R}$

1. $(z+a=a)\Rightarrow z=0$
2. $(b\ne 0\wedge u\cdot b=b)\Rightarrow u=1$
3. $a\cdot 0=0$

Proof

Drawing-2025-02-24-07.14.42.excalidraw