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In patent context, you put "the" only for the things which have been described previously.

I encountered "the first and a second insulating layers" which should have been as "the first insulating layer and a second insulating layer". However, I can't use the second phrase.

So, is it possible to write "a first and a second layer" ? and if so, should I put -s after the "layer" and put plural verb?

such as .. "a first and a second layer have~"

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    Context. CONTEXT! It's really important. I can imagine a situation where a first [something] is correct, but does that mean it was correct when you encountered it? Nope. – M.A.R. Apr 9 '15 at 6:46
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    Out of curiosity, why can't you write "the first insulating layer and a second insulating layer"? – snailcar Apr 9 '15 at 15:41
  • I would say a first and second layer. – user50333 Mar 1 '17 at 0:46
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I think you have a good grasp of the situation. "A first and a second layer" is certainly acceptable English, if a little stilted for normal conversation. In a technical context like a patent, it sounds more normal.

You would not pluralize "layer" if you're using the indefinite article "a".
Correct: "A first and a second layer"
Incorrect: "A first and a second layers"

You do use a plural verb form, because there is more than one layer.
Correct: "A first and a second layer are..."
Incorrect: "A first and a second is..."
(You will sometimes hear people say the "is" version in casual conversation, probably because it can sound a little strange to say something like "...a layer are...")

Here's a trick you can sometimes use to figure out what sounds best: try replacing some of the words with ones that have the same grammatical function, so you're not being distracted by their original meaning, and see what sounds more natural. For example, "A tall and a short boy..." is eating or are eating? Clearly it's are eating, so we know it has to be a plural form, even if that looks a little strange in the original context.

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