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I would like to know the difference between these prepositions when speaking about smth. what happens during a period of time or process.
For example behold these sentences (not mine):
1. Moreover she can break to pieces a good thing at such moments.
2. But it is impossible to work here without strict discipline in general and particulary at period of fulfilling arduous tasks.
It seems to me that there is something wrong with them. If you could, give please some correct examples of 'in' and 'at' when talking about a proces or a period of time.
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Number one sounds fine to me. Here we're talking about a moment in time, but it works for longer periods of time. We can discuss this at lunch time. (Some might argue that "during lunch time" is the correct word here. Probably so. Let's go for a coffee during intermission. "At" is more appropriate for brief moments. Should we applaud at the end of the first act?

Number two seems ungrammatical. Are you sure you copied it correctly? Something has to be missing. I don't really wish to speculate on what it's supposed to mean.
If you intend it as an example of "in," "in general" would have nothing to do with time.
If it's supposes to be "at periods," "during periods" would probably be a better choice.

Best wishes, - A.
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'In general' means 'generally' there. The question is whether or not it's possible to use 'at' instead of 'in', for example, before 'period': 'at period of'.
It may be possible, but my advice would be to use "at" only for points in time. "At the beginning of the war of 1812," "During the French Revolution," "In the 17th Century,"