This can be achieved using different worflows in FME. Since it is difficult to provide a complete solution without looking at some sample input data, I am not sure the example (trimline.fmw attached below) I provided here is the most suitable one for you. But I think it will help you to build a suitable solution.
In this workspace, I have created few line features to represent CS and HS lines as depicted in images you provided.
In the workflow, CS and HS lines are intersected to split them into line segments at intersections and forward them to the 'Line' input of PointOnLineOverlayer transformer. Start and End points of CS line have been created by extracting start and end coordinates forward to the 'Point' port of PointOnLineOverlayer transformer. Tester_2 transformer should contain a rule to filter CS line segments.
If Start and End points overlap with any CS line segment that segment should be removed to extract the trimmed line segment.
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