Pod pretpostavkom da je hemoglobin dobar, lekar će nastaviti.

It means assuming that, on the assumption that, or provided that.

It is a fairly formal expression. In everyday speech, people might also say:

• Ako je hemoglobin dobar, lekar će nastaviti. = If the hemoglobin is good, the doctor will continue.

So pod pretpostavkom da is a bit more official or medical/administrative in tone than a simple ako.

Because pretpostavkom is in the instrumental singular.

The noun is:

• pretpostavka = assumption

In the phrase pod pretpostavkom, the preposition pod takes the instrumental in this idiomatic meaning.

So:

• pretpostavka → nominative
• pretpostavkom → instrumental

This is something you often just learn as a set phrase:

• pod uslovom da = provided that
• pod pretpostavkom da = assuming that

Here da introduces a subordinate clause.

So the structure is:

• pod pretpostavkom = under the assumption
• da je hemoglobin dobar = that the hemoglobin is good

Together:

• pod pretpostavkom da je hemoglobin dobar = under the assumption that the hemoglobin is good

In Serbian, da very often introduces clauses after expressions like:

• mislim da... = I think that...
• važno je da... = it is important that...
• pod pretpostavkom da... = assuming that...

Because Serbian normally uses the present tense here: je = is.

The clause da je hemoglobin dobar literally means that the hemoglobin is good.

Even though the whole sentence refers to a future action (the doctor will continue), the condition itself is stated with the present form je.

So the logic is:

• if / assuming the hemoglobin is good now / at that point, the doctor will continue

Using bude would change the structure and sound different. For a learner, the safest choice here is to remember:

• pod pretpostavkom da je... = assuming that ... is ...

Because dobar is an adjective agreeing with hemoglobin.

• hemoglobin is masculine singular
• so the adjective must also be masculine singular
• therefore: dobar

Compare:

• hemoglobin je dobar = the hemoglobin is good
• krvna slika je dobra = the blood count is good
  (krvna slika is feminine, so dobra)
• rezultati su dobri = the results are good
  (rezultati is plural, so dobri)

It is in the nominative singular.

In the clause:

• hemoglobin je dobar

hemoglobin is the subject, and subjects are normally in the nominative.

So:

• hemoglobin = nominative singular
• dobar agrees with it in gender, number, and case

This is the future tense.

It is made with:

• the clitic će
• plus the infinitive nastaviti

So:

• će nastaviti = will continue

And:

• lekar će nastaviti = the doctor will continue

This is one of the standard ways to form the future in Serbian.

You may also see forms like:

• nastaviće

That is another way of writing/placing the future marker with the verb. Both are standard in many contexts, but će nastaviti is very clear and beginner-friendly.

Because će is a clitic, and Serbian clitics usually appear near the second position in their clause.

In the main clause:

• lekar će nastaviti

the clitic će comes after the first stressed element, here lekar.

This is very typical in Serbian:

• Ja ću doći.
• Oni će čekati.
• Lekar će nastaviti.

Clitic placement can feel unfamiliar to English speakers, because English does not organize short grammatical words in quite the same way.

Yes, to some extent.

This sentence begins with the condition:

• Pod pretpostavkom da je hemoglobin dobar, lekar će nastaviti.

That is very natural and slightly formal.

You could also say:

• Lekar će nastaviti pod pretpostavkom da je hemoglobin dobar.

Both are grammatical. The difference is mainly in focus and style:

• putting the conditional phrase first emphasizes the condition
• putting it later emphasizes the main action first

Serbian word order is more flexible than English, but not completely free.

Not exactly, though in many contexts they overlap.

• lekar = physician, medical doctor
• doktor can also mean doctor, but it can additionally mean someone with a doctorate, depending on context

In a medical sentence like this, both may be understood, but lekar is often the more precise general word for a physician.

So:

• lekar će nastaviti = the physician/doctor will continue

Yes, it is natural, especially in a formal, medical, or professional context.

A more everyday version would often be:

• Ako je hemoglobin dobar, lekar će nastaviti.

That means the same thing in a more direct, conversational way.

So the difference is mostly register:

• pod pretpostavkom da... = more formal, technical, official
• ako... = more common in everyday speech

Because the sentence starts with a subordinate clause / introductory conditional phrase, followed by the main clause.

So the structure is:

• Pod pretpostavkom da je hemoglobin dobar,
  introductory clause
• lekar će nastaviti.
  main clause

The comma helps separate those two parts.

In Serbian, commas are commonly used when a dependent clause comes before the main clause, just as in English:

• If the hemoglobin is good, the doctor will continue.

Yes, Serbian often omits subjects when they are understood from context.

So in the right context, you could simply say:

• Pod pretpostavkom da je hemoglobin dobar, će nastaviti.

However, that version sounds awkward because će is a clitic and usually wants a proper host before it. A more natural reduced version would be:

• Pod pretpostavkom da je hemoglobin dobar, nastaviće.

or, if the subject is already clear from context:

• Ako je hemoglobin dobar, nastaviće.

So yes, Serbian can drop subjects, but you also need to watch how that affects clitic placement.