Formal writing: replace "in fact" in a sentence

I am writing my PhD thesis, and I have a sentence that begins:

We will, in fact, prove a stronger condition:

Is there a more formal way to begin this sentence? Specifically, I am looking to remove "in fact" in favour of a more formal word or phrase.

I considered "Veritably, we will prove a stronger condition:", but it didn’t feel right.

I am also happy to be told "in fact" is formal enough.

EDIT:
The sentence that precedes this sentence reads:

Proving Theorem 1 hinges on demonstrating that…