LET P be the projective plane in [w4 obtained by capping off the boundary of an unknotted Miibius band in Iw3 x (0) with an unknotted disk in Iw3 x [O, co). Here we show that any smoothly imbedded projective plane in [w4 on which some projection Iw4+lR has three nondegenerate critical points is isotopic to P. The proof is based on a combinatorial solution to Problem 1.2B of [4]. In particular, if a band is attached to an unknot so that the result is an unknot, then the band is isotopic to the trivial half-twisted band. One consequence is that strongly invertible knots have property P (see [I]). Together with [23, this further implies that pretzel knots (indeed all symmetric knots) have property P. The solution of 1.2B uses the techniques of [S], [6] and [I], with careful distinction made between the two sides of the planar surfaces used in those arguments. Here is the philosophy: It was pointed out in [l] that the techniques of [S] and [6] were inadequate, because a certain type of semi-cycle ([l] Fig. 3) may arise, and yet cannot be used in the reduction process because there is no control over the side of the planar surface on which the interior of the semi-cycle lies. This was easily circumvented in [S], [6] because one of the planar surfaces involved was a punctured sphere. Here the planar surfaces are punctured disks, and it is difficult to detect whether a potential reducing disk is incident to just one side of the disk. In principle, simple bookkeeping should circumvent the problem. Yet there is a crucial step in the argument ([S, 6.31) at which all distinction between these sides seems hopelessly lost. In 6.9 an extremely weak analogue of [S, 6.31 is recovered, however, and prompts the study of “special paths”. These special paths are used, in three successive constructions of “multiflows”, to produce a semi-cycle whose interior must lie on a single side of a planar surface, and hence can be used to reduce the complexity of that surface. Roughly, the goal of the three successive stages is to remove from the interior of a multiflow first all sources, then all vertices, then all apexes. During the entire process, care is taken to ensure that at the end, all edges will have the same side on the interior.