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The part of the sentence "example explains that" is correct and usable in written English
You can use this part of the sentence whenever you are introducing a specific example to explain a previously stated idea. For example, "This example explains that there are many different ways to solve the same problem."
Exact(9)
The document, for example, explains that the attacks often involve a fair degree of coordination.
Google, for example, explains that its mission to organise the world's information is a continuation of its founders' research at Stanford University.
A wine wholesaler, for example, explains that, at $650 a bottle, his stock of Chateau Mouton Rothschild 1982 now costs less than in the United States.
One animation, for example, explains that, contrary to the common belief shared by both men and women in Cambodia, intimate partner violence is not normal and should be talked about.
A footnote to an edition of the Torah used in most Conservative synagogues until last year, for example, explains that Canaanites had to be conquered "because of the savage cruelty and foul licentiousness of their lives and cult".
This phenomenon is not limited to inclusiveness — the Dunning-Kruger effect, for example, explains that unskilled people are particularly prone to thinking they are more skilled than they are.
Similar(51)
Lawmakers in Colorado, for example, explained that cities had at times been overeager in their dealings with business interests.
Brown, for example, explained that his sexual identity opened up for him a diverse set of academic projects related to queer history and culture.
The lab might, for example, explain that "hatching asynchrony" — when one chick hatches before another — can produce one stronger chick that hogs resources and starves out the others.
Jack Straw, for example, explained that he was not talking about "real Romany Gypsies... who seem to mind their own business and don't cause trouble" but about "people who masquerade as travellers or Gypsies".
Mr Mulder, for example, explained that prints have become very expensive Picasso's poetic etching, "La Minatourmache" at his stand was priced at £1.2m ($1.9m)—that traditional print collectors have been priced out.
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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