Uniqueness of Inverse Operators Theorem and Proof

[Grand]

Homework Statement

Theorem
If, for given $$A$$, both operators $$A_l^{-1}$$ and $$A_r^{-1}$$ exist, they are unique and
$$A_l^{-1}=A_r^{-1}$$

The proof is rather straightforward, at least the first part of it:
$$A_l^{-1}A=I/\leftarrow A_r^{-1}$$
$$A_l^{-1}AA_r^{-1}=A_r^{-1} (1)$$

$$A_l^{-1}A=I/\rightarrow A_l^{-1}$$
$$A_l^{-1}AA_r^{-1}=A_l^{-1} (2)$$

Therefore
$$A_l^{-1}=A_r^{-1}$$

However, then they say that this proof holds for any pair of operators $$A_l^{-1}$$ and $$A_r^{-1}$$ (which I can't deny) and that eqs (1) and (2) ensure there exists only one such pair, which I can't understand. I would be very grateful if someone explains to me why is that.

 

[Dick]

If you've proved A_l^(-1)=A_r^(-1) then there's not really a pair. They are both the same operator, just call it A^(-1). Now suppose A had two different inverses, can you prove they are equal? It's really the same proof you just gave.