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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...

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    In fact is a cohesive device. It connects the current sentence to what has come before. So it is difficult to suggest an alternative or even comment on its necessity unless we know what comes before.
    – Shoe
    Commented Sep 5, 2020 at 8:31
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    Thanks for the clarification. In fact sounds fine to me in this context. Veritably sounds like something from a religious sermon and definitely not recommended here.
    – Shoe
    Commented Sep 5, 2020 at 9:43
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    Why do you want to sound as though you are writing in the 19th century? It only makes your thesis more difficult to read. If you want to sound more weighty, no-one who understands the subject will be fooled. Avoid slang, avoid humour, avoid poesy. Just use straightforward English. Spare the poor examiners having to wade through Shakespearean prose just to find out what you are talking about.
    – chasly - supports Monica
    Commented Sep 5, 2020 at 9:54
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    With your preceding sentence, 'in fact' here would fit in well with most of the academic articles I've read. 'Actually' works but sounds more conversational, so I'd discount it.
    – Edwin Ashworth
    Commented Sep 5, 2020 at 13:50
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    @chasly-reinstateMonica I'm less interested in sounding like I'm from the 19th century than I am in writing formally rather than conversationally. However, from the comments it seems that I'm over-thinking it and "in fact" is, in fact, formal enough.
    – David Poxon
    Commented Sep 5, 2020 at 23:11

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Essentially, you are asking for a word of higher register than the slightly over-used phrase in fact, which may also confuse readers by its reference to some undefined fact; one such word starts this answer.

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