Use of 'A' vs. 'The'?

Question

The rule states that:

Use (A) the first time you mention something, and (The) the second time you mention the same thing.

Kindly, consider the following example: (source: English for Starters 9)

((I'm going to tell you about a city called Tripoli, in the north of Lebanon. It is a city with lots of beautiful old houses.))

According to the rule, we should say:

((It is the city with lots of beautiful old houses))

because it is the second time we mention the word (city). Why did we use (a) instead of (the) in this situation?

Answer 1

In your question, it is not the number of times something is mentioned, rather

A

is used when being nonspecific, whereas

The

is used when being specific or pointing out uniqueness or utmost

I'm going to tell you about a city called Tripoli.
The city (that I just mentioned) has lots of beautiful houses.

"the city" is used since you are specifying the city you just mentioned in the previous sentence

This is a different pattern to

I'm going to tell you about a city called Tripoli.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.
It is the most famous and largest city in Lebanon.

Where there are many cities with "beautiful houses" and "architecture", but only one is the "*largest" and "most famous".

The ordering could be changed to

I'm going to tell you about a city called Tripoli.
It is the most famous and largest city in Lebanon.
It is a city with lots of beautiful houses.
It is also a city with pre-Roman and pre-Ottoman architecture.

A native speaker would not have a problem with either ordering.

Maybe this makes it clearer the difference in using "the" vs "a", but it is not dependent on the "ordering".

Answer 2

The rule is right, your example from the book is right.

Your interpretation is incorrect:

Your example in the book is correct. The first mention is A city called Tripoli. The second mention would then become The city, that specific city. BUT in their example, they continue with the idea of A city: it has not yet become The city. It is still a city among others in Lebanon.

Still referring to Tripoli.

If you say: It is a city with lots of beautiful homes that means there might be others like it in Lebanon.

The movement from A to THE has to refer to the same thing. If you start with A and keep A, that means the thing (city of Tripoli) is ONE of what could be others like it.