Proof

Suppose that a matrix $A$ is invertible. Then there exists $A^{-1}$ such that

$$A{A}^{-1}=A^{-1}A=I$$

where $I$ is an identity matrix.

By this equation, we can also consider that $A$ is the inverse matrix of $A^{-1}$.

Since the uniqueness of inverse matrix, we can conclude that

$$(A^{-1}{)}^{-1}=A$$