Uniqueness of Reciprocal and Multiplication Resulting in 0

Theorem

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

1. $(a\ne 0\wedge a\cdot b=1)\Rightarrow b=1/a$
2. $a\cdot b=0\Rightarrow (a=0\vee b=0)$

Proof

Drawing-2025-02-24-07.33.09.excalidraw