Αυτό είναι το αποδεικτικό στοιχείο.

A careful pronunciation is:

Αυτό είναι το αποδεικτικό στοιχείο
IPA: [afˈto ˈine to apoði(k)tiˈko stiˈçio]

A rough English-style guide:

• Αυτό → af-TO
• είναι → EE-ne
• το → to
• αποδεικτικό → a-po-thi-ktee-KO
  
  • the δ sounds like th in this
• στοιχείο → stee-HYO or stee-HEE-o
  
  • the χ is not an English sound; it is a harsher sound, somewhat like German ch

The written accent marks show the stressed syllable:

• αυτό
• είναι
• αποδεικτικό
• στοιχείο

Αυτό is the neuter singular form of this.

Greek has gender, so this changes form:

• αυτός = this (masculine)
• αυτή = this (feminine)
• αυτό = this (neuter)

Here, the thing being identified is το αποδεικτικό στοιχείο, and that noun phrase is neuter singular, so αυτό matches it.

This is very normal in Greek:

• Αυτό είναι το βιβλίο. = This is the book.
• Αυτή είναι η πόρτα. = This is the door.
• Αυτός είναι ο δάσκαλος. = This is the teacher.

είναι means is.

It is the 3rd person singular form of είμαι = to be.

So the structure is very straightforward:

• Αυτό = this
• είναι = is
• το αποδεικτικό στοιχείο = the evidence / the piece of evidence

Greek uses είναι just like English uses is in this kind of sentence.

Normally, no.

Greek often drops subject pronouns, but not the verb to be in a normal sentence like this.

So:

• Αυτό είναι το αποδεικτικό στοιχείο. = correct full sentence
• Αυτό το αποδεικτικό στοιχείο = not the same thing; this means this piece of evidence, not this is the piece of evidence

So if you want a complete sentence meaning This is the evidence, you need είναι.

το is the definite article: the.

So:

• το αποδεικτικό στοιχείο = the evidence / the piece of evidence
• ένα αποδεικτικό στοιχείο = a piece of evidence
• αποδεικτικό στοιχείο without an article can sound more general or less specific, depending on context

In this sentence, το shows that the speaker means a specific item of evidence, not just any evidence.

Grammatically, it is an adjective + a noun:

• αποδεικτικό = evidentiary / proving
• στοιχείο = element, item, piece

Together, αποδεικτικό στοιχείο is a very common expression meaning piece of evidence or simply evidence.

So even though English may use one word, Greek is using a noun phrase.

Because they are all neuter singular and they agree with each other.

In Greek, articles and adjectives must match the noun in:

• gender
• number
• case

Here:

• το = neuter singular article
• αποδεικτικό = neuter singular adjective
• στοιχείο = neuter singular noun

This agreement is one of the most important grammar patterns in Greek.

The phrase το αποδεικτικό στοιχείο is in the nominative here.

That is because after είμαι in a sentence like this, Greek normally uses a predicate nominative:

• Αυτό = subject
• είναι = verb
• το αποδεικτικό στοιχείο = predicate noun phrase

One small complication: in the neuter singular, the nominative and accusative forms often look the same. So you recognize the case here mainly from the sentence structure, not from a visible ending change.

This is a very good question.

In English, evidence is usually uncountable:

• we say some evidence
• not usually an evidence

Greek often expresses this idea with στοιχείο, which is countable:

• ένα αποδεικτικό στοιχείο = a piece of evidence
• δύο αποδεικτικά στοιχεία = two pieces of evidence

So Greek and English do not always package the idea in the same way. Greek often talks about an item/piece of evidence, even when English just says evidence.

Yes, Greek word order is more flexible than English, but the meaning or emphasis can change.

The neutral order here is:

• Αυτό είναι το αποδεικτικό στοιχείο.

You could also say:

• Το αποδεικτικό στοιχείο είναι αυτό.

That version puts more focus on αυτό, as if saying The piece of evidence is this one.

So the original sentence is the most neutral and natural choice if you simply want to identify something.

They are related, but not always identical.

• απόδειξη can mean proof
• in everyday Greek, απόδειξη also very commonly means receipt
• αποδεικτικό στοιχείο is more specific and often more formal: piece of evidence, especially in legal, investigative, or factual contexts

So in a sentence like this, το αποδεικτικό στοιχείο can sound more precise than just η απόδειξη.

Very roughly:

• απόδειξη = proof
• αποδεικτικό στοιχείο = evidentiary item / piece of evidence

Yes, but the meaning changes slightly.

• Αυτό είναι το αποδεικτικό στοιχείο.
  = This is the evidence / this is the piece of evidence
  It sounds specific, as if there is a particular item being identified.

• Αυτό είναι ένα αποδεικτικό στοιχείο.
  = This is a piece of evidence
  This sounds less definite: it is one item among others.

So the choice between το and ένα matters a lot.