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It seems to me that you can reason like this. Consider the boundary B of a small neighborhood N of $F$. Then this boundary B is connected if and only if F is 1-sided. Now, conisder the intersection of J with B. There are even number of intersections, since B is the boundary. So you can throw the part of J that does not belong to the neighbohood N an close it to a connected curve by segments in B (we assumed that B is connected).