grammar - How do you explain the usage of "with"?


I'm looking for a logical explanation of how we use the preposition "with" in the following context:



The company took the first place with their competition a distant second.


North America had the greatest number of travellers over the 5-year period with Central and Eastern Europe showing a similar pattern.



I suppose my main question is how do I explain the phrase (it's not a clause) that follows the "with"? and how do I explain the function of "with" in these sentences?




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