هذا إثبات الملكية.

هذا means this for a masculine singular noun.

In هذا إثبات الملكية, it points to the thing being referred to: This is proof of ownership.

A useful pattern to remember is:

• هذا كتاب = This is a book
• هذه سيارة = This is a car

Here, هذا is used because إثبات is a masculine noun.

Because إثبات is grammatically masculine.

In Arabic, demonstratives must agree with the noun in gender:

• هذا = this (masculine)
• هذه = this (feminine)

So since إثبات is masculine, Arabic uses هذا:

• هذا إثبات = This is proof/documentation
• not هذه إثبات

This is about grammatical gender, not biological sex.

In Arabic, the verb to be is usually not expressed in the present tense.

So:

• هذا إثبات الملكية literally = This proof of ownership but naturally means = This is proof of ownership

This is very common in Arabic nominal sentences.

Compare:

• البيت كبير = The house is big
• هذا كتاب = This is a book

But in the past or future, Arabic does use words equivalent to was / will be.

إثبات comes from a root related to confirming, proving, or establishing something.

Depending on context, it can mean:

• proof
• verification
• confirmation
• documentation

In the phrase إثبات الملكية, it means something like:

• proof of ownership
• ownership documentation

So this sentence could refer to a document or evidence showing that someone owns something.

الملكية means ownership or possession.

The الـ at the beginning is the definite article, meaning the.

So:

• ملكية = ownership
• الملكية = the ownership

In this phrase, إثبات الملكية means proof of ownership.

Even though English often says proof of ownership without the, Arabic commonly uses الـ in this kind of construction.

Yes, it is effectively an iḍāfa-style relationship meaning proof of ownership.

The structure is:

• إثبات = proof
• الملكية = the ownership

Together they express a close relationship: proof of ownership.

A very important point for learners: In a straightforward iḍāfa, the first noun usually does not take الـ, while the second noun may be definite. That is exactly what we have here:

• إثبات الملكية not
• الإثبات الملكية

This same pattern appears in many common expressions:

• باب البيت = the door of the house
• مدير المدرسة = the principal of the school

A simple pronunciation guide is:

• هذا = haadha
• إثبات = ithbaat
• الملكية = al-milkiyya

So the full sentence sounds roughly like:

haadha ithbaat al-milkiyya

A few notes:

• ث is like th in think
• إثبات has a long aa sound at the end: ith-baat
• الملكية has kiy-ya near the end

If you want a smoother full pronunciation: haadha ithbaatul-milkiyya can be heard in careful reading with case endings, though in everyday spoken MSA those endings are often not fully pronounced.

In fully vowelled, formal Arabic, yes, the grammatical endings would typically be:

• هذا إثباتُ الملكيةِ

Why?

• إثباتُ is the predicate of the sentence, so it is normally nominative: -u
• الملكيةِ is the second term in the iḍāfa construction, so it is genitive: -i

However, in normal printed Arabic, these short vowel endings are usually not written, so you commonly see:

• هذا إثبات الملكية

Both refer to the same sentence; the second is simply the standard unvowelled spelling.

It most naturally means This is proof of ownership.

Because إثبات does not have الـ, it is not explicitly the proof.

So:

• هذا إثبات الملكية = This is proof of ownership
• هذا الإثباتُ للملكية or another more specific phrasing could be used for This is the proof of ownership, depending on context

That said, translation depends on situation, and English may sometimes add the if it sounds more natural in context. But grammatically, the Arabic here is more naturally understood as proof of ownership, not specifically the proof.