Does the agency of someone appointed by the guardian's agent continue after the termination of the guardian's agency?

It is permissible for an agent to appoint another agent, and the first principal may dismiss the second agent. If the agency of the first agent is terminated, the agency of the second agent is consequently terminated. Scholars differ regarding the validity of the first agent dismissing the second agent, based on the phrasing used by the first principal: if the principal said to his agent, "Appoint on your behalf," or if it necessitated designation, then it is valid, and the second agent would be an agent for the first, thus his dismissal would be revoked by the death of the first agent. However, if the principal said, "Appoint on my behalf," or used an unrestricted phrase, then it is valid, and the second agent would be an agent for the principal, and therefore would not be dismissed by the dismissal of the first agent nor by his death.