تنتسب إلى المكتبة بشرط أن يكون إثبات العنوان في استمارة العضوية.

Because the bare written form تنتسب can match more than one subject in Arabic when short vowels are not written.

It can be:

• you (masculine singular): تنتسبُ
• she: تنتسبُ

So the meaning comes from context. In this sentence, it sounds like an instruction or rule addressed to the reader, so it is understood as you join / you become a member.

If the sentence were directly addressing one female, you would usually expect تنتسبين.

Because Arabic verbs usually include the subject inside the verb form itself.

So تنتسب already tells you the subject. Arabic does not need to add أنت unless there is a reason to emphasize it, contrast it, or clarify it.

So:

• تنتسب إلى المكتبة = You join the library
• أنت تنتسب إلى المكتبة = You join the library, with extra emphasis

تنتسب comes from the verb انتسب إلى.

In Modern Standard Arabic, انتسب إلى commonly means:

• to join
• to become affiliated with
• to become a member of

So with المكتبة, it means joining or becoming a member of the library.

It is a little more formal than everyday spoken verbs like يسجّل in some contexts.

Because انتسب normally takes the preposition إلى.

So Arabic says:

• انتسب إلى المكتبة
• انتسب إلى الجامعة
• انتسب إلى النقابة

This is just the standard pattern of the verb. It is best to learn انتسب إلى as one unit.

بشرط أن is a very common expression meaning:

• provided that
• on condition that
• as long as

Literally:

• بـ = with
• شرط = condition

So بشرط أن introduces a condition followed by a clause.

Example:

• تدخل بشرط أن تلتزم بالقوانين
  You may enter provided that you follow the rules.

After بشرط أن, Arabic normally uses a verb clause introduced by أن.

Here, يكون is used to express be:

• بشرط أن يكون... = provided that ... be / is

Arabic often needs يكون in places where English may simply say that there is or just use be without thinking much about it.

Also, after أن, the imperfect verb is in the subjunctive. So in full vowelled Arabic it would be يكونَ. In normal writing, that final short vowel is usually not written, so you just see يكون.

Because it agrees with إثبات, which is a masculine singular noun.

The idea is that proof of address is the thing that is being talked about, so the verb appears in masculine singular form:

• يكون إثبات العنوان...

If the noun were feminine singular, you would often see a feminine form instead.

Literally, إثبات means proof or verification, and العنوان means address.

So إثبات العنوان means:

• proof of address
• or more literally, verification of the address

In administrative Arabic, إثبات often refers to some kind of document or evidence, not only the abstract act of proving.

Because this is an iḍāfa construction, often called a possessive or of construction.

Structure:

• إثبات = proof
• العنوان = the address

Together:

• إثبات العنوان = proof of the address / proof of address

In an iḍāfa:

• the first noun usually does not take ال
• the second noun determines whether the whole phrase is definite

Since العنوان is definite, the whole phrase إثبات العنوان is definite too.

The same thing happens in استمارة العضوية.

It is also an iḍāfa construction:

• استمارة = form
• العضوية = membership

So:

• استمارة العضوية = membership form
• or membership application form, depending on context

Again, the first noun has no ال, and the second noun makes the whole phrase definite.

Because Arabic and English do not always use prepositions the same way.

Arabic commonly uses في in places where English might say:

• in the form
• on the form
• in the application

So في استمارة العضوية is normal Arabic, even if English would often prefer on the membership form.

This is a good reminder not to translate prepositions word-for-word.

Formally, تنتسب is in the imperfect, which can cover:

• present
• future
• habitual meaning
• general rules

In this sentence, it reads like a general rule or condition, not necessarily something happening right now.

So it is closer to:

• You can join the library provided that...
• or You join the library provided that...

Arabic often uses the imperfect for this kind of rule-like statement.

Yes. That is very common.

Although إثبات is originally a verbal noun, in administrative or formal language it often refers to actual proof or supporting documentation.

So إثبات العنوان can naturally mean:

• proof of address
• address verification
• a document showing your address

That kind of noun usage is very common in official Modern Standard Arabic.