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If you are talking about a gradient along a path, cutting and rearranging the paths may produce some undesired results and things can get pretty complicated.
I would probably just use a compound clipping path to do it. See sample Illustrator file below.
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Revert the path direction?
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Thanks Monika, but that changes only the gradiant.
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This is how I understood your request. So that's not what you want?
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Non actually, I want the darker line behind the brighter one, just where these parts overlap
 
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You'll have to cut the path.
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Dear Monika,
as Kurt mentions it below the gradiant will be affectet by clipping it.
Thanks for your support!
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If you are talking about a gradient along a path, cutting and rearranging the paths may produce some undesired results and things can get pretty complicated.
I would probably just use a compound clipping path to do it. See sample Illustrator file below.
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Dear Kurt,
thanks for the file! That works and I will adapt your sollution to my file. Thank you!
Nevertheless, I had thought that there was a tool for changing the order of intersecting elements. Somehow a kind of lasso tool is buzzing around in my head. But maybe that was in a different programme.
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Recent Illustrator versions have the Intertwine tool, but it doesn't work in your specific case. It is useful in many other cases.
Behind the scenes, the Intertwine tool just uses clipping path constructions, by the way.
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Thanks!
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Intertwine does not work with one object.