Simplifying intersections of surfaces.

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