Tämä todistaa suunnitelman toimivan.

Because this sentence uses a compact Finnish structure meaning that the plan works.

In that structure, suunnitelman is the logical subject of toimivan, and it appears in the genitive form:

• suunnitelma = plan
• suunnitelman = genitive form

So suunnitelman toimivan means roughly that the plan works / is working.

A useful comparison is:

• Tiedän hänen tulevan. = I know that he/she is coming.

There too, the subject of the shorter embedded clause is in the genitive.

It comes from the verb toimia, which means to work, to function.

Here toimivan is the present active participle in the form used in this construction. The basic participle is toimiva, which can often mean working, functional, effective.

But in this sentence, toimivan is not just describing suunnitelman like an ordinary adjective. Together, suunnitelman toimivan expresses a whole idea:

• that the plan works

So it is more verbal than adjectival here.

Yes, the meaning is essentially the same.

• Tämä todistaa suunnitelman toimivan.
• Tämä todistaa, että suunnitelma toimii.

The second version uses a full että-clause and is often easier for learners to understand.

The first version is more compact and can sound a bit more written, formal, or stylistically tight.

Because toimii is a normal finite verb form, so it belongs in a full clause:

• että suunnitelma toimii

In the shorter construction, Finnish replaces that full clause with a participial structure:

• suunnitelman toimivan

So:

• toimii = used in a full clause
• toimivan = used in this compressed clause-equivalent structure

Because this pattern uses forms that happen to look similar:

• suunnitelman is the genitive of suunnitelma
• toimivan is the participial form used in this construction

So the matching -n endings are not a coincidence, but they are not doing exactly the same job either. Finnish is using a fixed grammatical pattern here.

The best way to think about it is:

• suunnitelman is the logical subject of toimivan
• the whole phrase suunnitelman toimivan is what todistaa takes as its object-like complement

So Finnish is not really saying:

• This proves the plan

It is saying:

• This proves that the plan works

That whole final idea is packaged into suunnitelman toimivan.

Not in the same way.

Compare these two:

• toimiva suunnitelma = a working / effective plan
• suunnitelman toimivan = that the plan works

In toimiva suunnitelma, toimiva behaves like an adjective modifying the noun.

In suunnitelman toimivan, the participle is part of a verbal structure expressing an embedded statement.

So even though the forms are related, the grammar is different.

No. That changes the structure completely.

• toimiva suunnitelma = a working / effective plan
• toimivan suunnitelman = a case form of working / effective plan

So Tämä todistaa toimivan suunnitelman would be understood as something involving a working plan as a noun phrase, not that the plan works.

The original sentence needs:

• suunnitelman toimivan

That order is important.

It is often called a clause-equivalent construction in English descriptions of Finnish grammar. In Finnish grammar, it is commonly discussed as a type of lauseenvastike.

In practical terms, it is a compact way to express something that English would usually put in a full that-clause:

• että suunnitelma toimii
• suunnitelman toimivan

For a learner, the easiest approach is often:

1. recognize it as a shortened that-clause
2. identify the noun in the genitive as the logical subject
3. identify the participle as the verb idea

Yes, but the full että-clause is often simpler and more common in casual speech:

• Tämä todistaa, että suunnitelma toimii.

The participial version:

• Tämä todistaa suunnitelman toimivan.

is very normal too, especially in written Finnish, formal style, reporting, argumentation, and contexts where a more compact expression is preferred.