Eers nadat die lugdiens die aanspraak verwerp het, begin die bemiddeling.

Eers nadat means only after.

• nadat = after
• eers adds the idea of not before then / only then

So Eers nadat die lugdiens die aanspraak verwerp het means Only after the airline has rejected the claim.

Without eers, nadat would just mean after, but eers makes the timing more restrictive.

Because Afrikaans uses verb-first inversion when something other than the subject comes at the start of the sentence.

The sentence begins with the time clause:

Eers nadat die lugdiens die aanspraak verwerp het, ...

After that opening element, the finite verb of the main clause comes next:

• begin = finite verb
• die bemiddeling = subject

So the main clause is:

begin die bemiddeling

This is very common in Afrikaans. Compare:

• Die bemiddeling begin. = The mediation begins.
• Môre begin die bemiddeling. = Tomorrow the mediation begins.
• Eers daarna begin die bemiddeling. = Only after that does the mediation begin.

Because in Afrikaans subordinate clauses, the verbs normally go to the end.

Here the subordinate clause is:

nadat die lugdiens die aanspraak verwerp het

This contains a perfect construction:

• verwerp = rejected
• het = has

In a main clause, you would normally get:

Die lugdiens het die aanspraak verwerp.

But after nadat, it becomes:

... nadat die lugdiens die aanspraak verwerp het.

So the verb elements move to the end of the subordinate clause.

In a normal main clause, Afrikaans usually puts the auxiliary in second position:

• Die lugdiens het die aanspraak verwerp.

But in a subordinate clause introduced by a word like nadat, the verb cluster goes to the end:

• ... nadat die lugdiens die aanspraak verwerp het

So verwerp het is not random word order; it is the expected subordinate-clause order.

Lugdiens means airline.

It is made from:

• lug = air
• diens = service

So literally it is something like air service, but the normal English translation here is airline.

In this sentence, aanspraak means claim.

So:

• die aanspraak verwerp = reject the claim

Be aware that aanspraak can have slightly different meanings in other contexts, such as a right, entitlement, or claim to something. But here, in a legal or complaint-related context, claim is the natural meaning.

Yes. Bemiddeling is the noun form related to the verb bemiddel.

• bemiddel = to mediate
• bemiddeling = mediation

So:

• begin die bemiddeling = the mediation begins

This -ing ending often forms nouns in Afrikaans, much like English -ing or -tion in some cases, though the match is not always exact.

Afrikaans does not usually use a separate helping verb like English do/does for this kind of sentence.

English says:

• Only after ..., does the mediation begin.

Afrikaans simply uses the main verb:

• ... begin die bemiddeling.

So the idea expressed by English does begin is carried by normal Afrikaans word order, not by an extra verb.

The comma separates the opening subordinate clause from the main clause.

Structure:

• Eers nadat die lugdiens die aanspraak verwerp het, = subordinate clause
• begin die bemiddeling. = main clause

This helps the reader see where the first clause ends and where the main statement begins.

Yes, nadat is a subordinating conjunction, so it introduces a subordinate clause.

That means:

• it is followed by a full clause
• the verb or verb cluster goes to the end of that clause

Example:

• Nadat hy geëet het, vertrek hy. = After he has eaten, he leaves.
• Nadat die lugdiens die aanspraak verwerp het, begin die bemiddeling.

So if you use nadat, you should expect subordinate-clause word order.

Yes, but the meaning would change slightly.

• Nadat die lugdiens die aanspraak verwerp het, begin die bemiddeling.
  = After the airline has rejected the claim, the mediation begins.

• Eers nadat die lugdiens die aanspraak verwerp het, begin die bemiddeling.
  = Only after the airline has rejected the claim does the mediation begin.

So eers adds emphasis that the mediation will not begin before that point.

The main clause is in the present tense:

• begin die bemiddeling = the mediation begins

But the subordinate clause uses the perfect:

• die lugdiens ... verwerp het = the airline has rejected

So the sentence describes a sequence:

1. the airline rejects the claim
2. then the mediation begins

This is why English often translates it as Only after the airline has rejected the claim does the mediation begin.