The with the

Hi,

Generally, what starts with an article "a" or "an" will be followed with the same article like this:

An apple is a kind of fruit.

I think the same can be said of the article "the" like this:

The name is the word that you are being called as/with.

But how about here? Do you really need the article "the" here?

The two names are the ??? proper names.

I think, the sentence without "the," "The two names are proper names", makes good sense.

Also, how would you punctuate the colored part?