Sin el justificante, mi hermana no pudo demostrar que ya había pagado.

In Spain, justificante commonly means some kind of proof or supporting document, especially a receipt, proof of payment, or document that shows something was done.

In this sentence, el justificante is most naturally understood as the receipt / proof of payment.

A useful Spain-specific note:

• In Spain, justificante is very common in everyday and administrative language.
• In many Latin American countries, learners may hear comprobante more often in similar contexts.

Both are possible, but they are not exactly the same.

• sin el justificante = without the receipt / without that specific proof document
• sin justificante = without any receipt / without proof

Using el makes it sound like a particular receipt or document is being referred to, probably one that was expected in that situation.

So:

• Sin el justificante... points to a specific document.
• Sin justificante... is more general.

No pudo is the preterite of poder, and here it presents the failure as a completed event in the past.

So the sentence means something like:

• she was unable to prove it at that moment / on that occasion

By contrast, no podía demostrar would usually suggest:

• an ongoing inability,
• a general situation,
• or background information rather than a single completed event.

Compare:

• No pudo demostrarlo.
  She couldn’t prove it. / She failed to prove it.

• No podía demostrarlo.
  She wasn’t able to prove it / she couldn’t prove it, with more of a continuous or descriptive feel.

In this sentence, the speaker is describing a specific past situation, so no pudo fits better.

Because poder is a modal verb. In Spanish, modal verbs such as poder, deber, and querer are commonly followed by an infinitive.

So:

• pudo demostrar = was able to prove / managed to prove
• quiere comer = wants to eat
• debe salir = must leave

You do not conjugate the second verb here. Only poder is conjugated:

• pudo = past tense of poder
• demostrar stays in the infinitive

Había pagado is the pluperfect (also called the past perfect):
haber in the imperfect + past participle

It is used to show that one past action happened before another past action.

Here the two past events are:

1. she had already paid
2. later, she could not prove it

So había pagado makes the time relationship very clear: the paying happened earlier.

Why not the others?

• pagó = she paid
  This is simple preterite, and it would not show the “earlier than another past action” relationship as clearly.

• ha pagado = she has paid
  This is the present perfect, which normally connects the action to the present. That does not fit as well here because the main narrative is already in the past: no pudo demostrar.

Ya means already here.

It emphasizes that the payment had happened before the moment when she needed to prove it.

So:

• había pagado = she had paid
• ya había pagado = she had already paid

It often appears before the verb phrase:

• ya había pagado

But you may also hear:

• había pagado ya

Both are possible, though ya había pagado is very natural and common.

Because que introduces a subordinate clause:
que ya había pagado = that she had already paid

The structure is:

• demostrar + que + clause

Examples:

• Demostró que tenía razón.
  She proved that she was right.
• No pudo demostrar que había pagado.
  She couldn’t prove that she had paid.

In English, that is sometimes optional, but in Spanish que is normally required in this structure.

Yes. A few common alternatives are:

• probar
  very common for to prove
• acreditar
  more formal, often used in administrative or legal contexts
• justificar
  sometimes possible, but it often means to justify or to provide supporting evidence, so it is not always the exact same thing

Examples:

• no pudo probar que ya había pagado
• no pudo acreditar que ya había pagado

In Spain, with documents and bureaucracy, acreditar and justificante can sound especially natural in formal settings.

Spanish often prefers a noun instead of repeating a subject pronoun, especially when it helps keep the sentence clear or natural.

So:

• mi hermana no pudo demostrar... sounds perfectly natural.
• ella no pudo demostrar... is also possible, but it depends on context.

Spanish usually omits subject pronouns unless they are needed for:

• emphasis,
• contrast,
• or clarity.

For example:

• Mi hermana no pudo demostrarlo, pero yo sí.
  Here yo is used for contrast.

In standard Spanish, possessives like mi, tu, su, nuestro usually go directly before the noun without an article:

• mi hermana
• tu coche
• su casa

So la mi hermana is not standard modern Spanish in this context.

You may sometimes see article + possessive in older language, regional uses, or special emphatic structures, but for normal modern Spanish, mi hermana is the correct form.

The word order can be changed somewhat.

The original sentence:

• Sin el justificante, mi hermana no pudo demostrar que ya había pagado.

A very natural alternative is:

• Mi hermana no pudo demostrar que ya había pagado sin el justificante.

But the meaning can shift slightly depending on what you want to emphasize.

Starting with Sin el justificante puts the missing receipt in focus first. It sets the scene:

• Without the receipt, ...

That kind of fronting is very common in Spanish.

Yes, if the subject is already clear from context.

You could say:

• Sin el justificante, no pudo demostrar que ya había pagado.

Since pudo is third person singular, the subject could be:

• he
• she
• you (formal)

So if context already tells us it is the sister, the noun can be omitted. Spanish does this a lot because the verb ending often gives enough information.

No. Justificante is broader than just payment.

It can refer to any document that serves as proof, for example:

• proof of payment
• a medical note
• proof of attendance
• a document explaining an absence

But in this sentence, because of había pagado, the most logical meaning is proof of payment or receipt.

Yes. These are both possible:

• que ya había pagado
• que había pagado ya

Both mean the same thing. The difference is mainly rhythm and emphasis.

• ya había pagado is the more neutral and common order.
• había pagado ya can sound a bit more emphatic or stylistically marked, depending on context.

So the original version is a very natural choice.