Suppose μ and v are arcs whose union is an arc α and whose intersection is an arc β. Suppose further that μ and v, respectively, lie in the boundaries of the disks M and N. It is shown that either α lies on the boundary of a disk or μ and v, respectively, lie in the boundaries of disks M′ and N′ which meet only in the arc β. The construction is basic to recent results exploring new questions formerly considered only in the piecewise linear category.
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