Račun s markom i modelom dokaz je vlasništva.

Because the preposition s meaning with normally takes the instrumental case in Croatian.

So:

• marka → markom
• model → modelom

That is why you get:

• s markom = with the brand
• s modelom = with the model

When two nouns are joined by i (and), both stay in the same case:

• s markom i modelom = with the brand and model

This is very common in Croatian:

• s prijateljem = with a friend
• s knjigom = with a book

Marka is a feminine noun, and here it is in the instrumental singular because it follows s.

A basic pattern for many feminine nouns ending in -a is:

• nominative: marka
• genitive: marke
• dative: marki
• accusative: marku
• vocative: marko
• locative: marki
• instrumental: markom

So in this sentence:

• s markom = with the brand

Because model is a masculine noun, and after s it also has to be in the instrumental singular.

For many masculine nouns, the instrumental singular ends in -om or -em:

• nominative: model
• instrumental: modelom

So:

• s modelom = with the model

Vlasništva is in the genitive singular.

The noun dokaz often takes a genitive complement, much like English proof of ...:

• dokaz vlasništva = proof of ownership
• dokaz identiteta = proof of identity

So the structure is:

• Račun s markom i modelom = the receipt with the brand and model
• je = is
• dokaz vlasništva = proof of ownership

This is a very standard Croatian pattern: noun + genitive.

Croatian normally does not use articles like English a/an/the.

So račun can mean:

• a receipt
• the receipt

The exact meaning depends on context.

Likewise:

• dokaz vlasništva can mean proof of ownership
• the proof of ownership, depending on context

English learners often look for an article, but Croatian usually leaves it out entirely.

In this sentence, račun means receipt.

But račun can have several meanings depending on context, for example:

• receipt
• bill
• invoice
• calculation
• account (in some contexts)

Here, because the sentence talks about brand and model and proof of ownership, receipt is the natural meaning.

Je is the 3rd person singular of biti (to be) and means is.

The sentence is structured like this:

• Račun s markom i modelom = subject
• je = is
• dokaz vlasništva = predicate noun phrase

So literally:

• Receipt with brand and model is proof of ownership.

Croatian word order is more flexible than English, but this order is very normal and natural.

In a full standard sentence like this, je is normally not omitted.

So:

• Račun s markom i modelom je dokaz vlasništva. = correct

In headlines, notes, labels, or very compressed style, Croatian may drop forms of to be, but in ordinary complete writing and speech, you usually keep je.

Because dokaz here is in the nominative, not the instrumental.

This sentence is an equational sentence:

• X je Y = X is Y

Both the subject and the predicate noun are commonly in the nominative in Croatian:

• Račun ... je dokaz ...

So:

• račun = nominative
• dokaz = nominative

You would use dokazom only in a different grammatical role, for example:

• To služi kao dokazom would actually sound wrong in standard Croatian
• more natural: To služi kao dokaz = That serves as proof

So in this sentence, dokaz must stay nominative.

Yes, s and sa both mean with, and both take the instrumental.

Usually:

• s is the basic form
• sa is used for easier pronunciation before certain consonant clusters or sounds

Examples:

• s prijateljem
• sa sestrom
• sa mnom

In your sentence, s markom i modelom is the normal choice.
Using sa markom i modelom would sound less natural here.

Here marka means brand or make, especially for a product or device.

So marka i model is a very common pair meaning:

• brand and model
• make and model

In practical English, make and model is often the best translation if you are talking about an object like a phone, laptop, car, or appliance.

Yes, a version with o vlasništvu may be possible in some contexts, but dokaz vlasništva is the more straightforward and idiomatic phrasing here.

Compare:

• dokaz vlasništva = proof of ownership
• dokaz o vlasništvu = evidence/document about ownership

The first one is the cleaner, more usual expression for proof of ownership, especially in official or practical usage.

Very literal:

• račun = receipt
• s = with
• markom = brand/make
• i = and
• modelom = model

So:

• receipt with the brand and model

The idea is that the receipt includes or shows that information, and because of that, it can serve as proof that you own the item.

A useful breakdown is:

• Račun — nominative singular masculine noun
• s markom i modelom — prepositional phrase with instrumental
• je — 3rd person singular of biti
• dokaz — nominative singular masculine noun
• vlasništva — genitive singular after dokaz

So the pattern is:

[subject] + [with + instrumental] + je + [predicate noun + genitive complement]

That makes this sentence a good example of two important Croatian structures at once:

1. s + instrumental
2. noun + genitive for meanings like proof of ...