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The phrase "consistent with the fact" is correct and can be used in written English.
It is typically used to indicate that something agrees or is in line with a known or proven fact or truth. Example: These results are consistent with the fact that exercise improves overall health.
Exact(60)
This is consistent with the fact that human sleep consists of repeating sleep units.
This is hardly consistent with the fact that these officials have offered another building.
"I think that is a position which is not consistent with the fact," Mr. Romney said.
And that, again, is a smoother reconstruction consistent with the fact that the triangles are smoother than the rectangles.
A second is subtracted off to keep consistent with the fact that the state machine is a second behind reality.
This is consistent with the fact that memory shows pronounced deficits in SZ, with working memory primarily affected22,47.
We uncover little effect of subsidies on academic researchers, consistent with the fact that their incentives are unaffected.
This is consistent with the fact that the nozzle ablation was mainly diffusion-limited when the wall temperature was high.
This is consistent with the fact that these effects are primarily the result of the interaction between inclusions.
This is consistent with the fact that our sample over-represents large employment websites.
This is consistent with the fact that long-chain alkanes are prone towards cracking.
More suggestions(15)
coherence with the fact
consonant with the fact
consistent with the way
consistent with the set
consistent with the situation
consistent with the experiment
consistent with the measurement
consistent with the disease
consistent with the morphology
consistent with the anthrax
consistent with the approach
consistent with the government
consistent with the site
consistent with the formulation
consistent with the literature
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com