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The phrase "there exist two" is grammatically correct and can be used in written English.
It is commonly used in mathematical and scientific writing to denote the existence of something in a set or group. Here are a few examples of how it can be used: 1. "In this experiment, we have shown that there exist two distinct types of reactions when the element is exposed to high temperatures." 2. "The research team concluded that there exist two possible theories to explain the phenomenon, but further study is needed to determine which one is correct." 3. "There exist two main methods of data collection in this field of study - surveys and interviews." 4. "According to the survey results, there exist two dominant trends in consumer behavior: price sensitivity and brand loyalty." Overall, the phrase "there exist two" is a valid and concise way to express the existence of two things in written English.
Exact(60)
In the relevant literature there exist two dominant approaches.
If, and, then there exist two positive constants, such that.
For TD-learning there exist two different types of convergence.
Basically, there exist two approaches.
There exist two main semantics for Esterel.
Then there exist two vertices such that.
There exist two constants such that and,.
There exist two constants such that.
For with there exist two positive solutions.
There exist two kinds of proton pathways.
There exist two constants such that (2.5).
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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