What reason did Sherlock Holmes put forward to prove that C.C.H stood for Charing Cross Hospital, and that Dr. Mortimer had moved from town to country?

From the text:

"I would suggest, for example, that a presentation to a doctor is more likely to come from a hospital than from a hunt, and that when the initials ‘C.C.’ are placed before that hospital the words ‘Charing Cross’ very naturally suggest themselves.’"

"We know there has been a presentation. We believe there has been a change from a town hospital to a country practice. Is it, then, stretching our inference too far to say that the presentation was on the occasion of the change?"

Watson walks into Holmes's breakfast-room, where Sherlock Holmes is having breakfast. Watson examines a walking stick which a visitor, James Mortimer, had left behind the night before, after finding nobody there to receive him. Mortimer's name is engraved into the stick.

Though his back is turned to Watson, Holmes sees his friend through the reflection in his coffee-pot. He surprises Watson by addressing him, and then asks him to deduce the character of James Mortimer from his stick.

Based on the stick, Watson believes that Dr. Mortimer is an elderly, well-respected doctor who lives in the country. He further deduces that Mortimer has received this stick as a gift from a hunting club.

Holmes initially compliments Watson's detective skills, but then clarifies that he is only complimenting the way that Watson has stimulated his own thought process. Holmes examines the stick himself, and concludes that Mortimer received the stick as a gift from a hospital, rather than from a hunting club. He deduces that Mortimer was a student at this hospital, not a physician, and that he must therefore be young, not old. Further, he believes that Mortimer has withdrawn from a town hospital to begin his own practice in the country. He adds that Mortimer must be absent-minded, amiable, unambitious and a dog owner.

Astonished, Watson looks in his Medical Dictionary for public information about Mortimer. The book confirms that Mortimer is a young man who studied as Charing Cross Hospital.