The conclusion of Theorem HMIP is $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{\adjoint{A}\vect{y}}$. Use the same hypotheses, and prove the similar conclusion: $\innerproduct{\vect{x}}{A\vect{y}}=\innerproduct{\adjoint{A}\vect{x}}{\vect{y}}$. Two different approaches can be based on an application of Theorem HMIP. The first uses Theorem AA, while a second uses Theorem IPAC. Can you provide two proofs?