Proof of Uniqueness of the Matrix Inverse

Each result cited is universally quantified over the data in its own statement. The rules of Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes are used with $m=n=p=q$.

Assume $AB=I_{n}$ and $CA=I_{n}$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §identity (with $B$, respectively $C$, in place of $A$), $I_{n}B=B$ and $CI_{n}=C$; by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §associative (with $C$, $A$, $B$ in place of $A$, $B$, $C$), $(CA)B=C(AB)$. Hence

$$B=I_{n}B=(CA)B=C(AB)=CI_{n}=C.$$