Original between-copy exchanges contained in normal products are vertical

Use the actual small rectangular switching neighborhood of [F1]. Inside each tetrahedron a piece of $A$ is a rectangle with its two frontier sides on the full-resolution sheets and its other two sides in the two distinct faces crossed by that original double arc. Its two frontier sides lie respectively on $D_0$ and $D_1$: these are the two entire frontier components stipulated in the Statement, so following either component cannot change its full-resolution component. The rectangle's interior misses $F$ and all original edges. Different such pieces have disjoint interiors because the selected exchange surface and its switching neighborhood are embedded.

Each such open rectangle lies in one tetrahedral prism of $P$. Indeed the interiors of those prisms are the components between their horizontal normal sheets inside that tetrahedron. Passing from one to another inside the tetrahedron would cross a horizontal $D_0$ or $D_1$ sheet or an original edge; the rectangle's interior crosses neither. Its two frontier sides consequently lie on the two horizontal ends of that one prism. They cannot be different normal pieces on the same horizontal end, since one side belongs to $D_0$ and the other to $D_1$. Thus they lie on the corresponding disk pieces bounding the prism. Their endpoints are on the same two distinct original faces, by step 1.1. Similarly, in a face strip each side of the exchange gives a proper arc joining the two horizontal arcs: it cannot leave the strip across an edge interval, since it misses the original edges. In the proper-disk case the remaining boundary of $A$ lies in $\mathrm{\partial }M$ and hence in the prescribed vertical boundary of $P$. These are all the local rectangle and incidence conditions of [F2], now obtained from the actual diagram and containment, rather than imposed on an unrelated spanning surface.

Apply [F2] to these finite rectangle and face-strip families. Its relative disk-region maps on the face strips agree at the original edge intervals, and its proper-disk cutting construction extends them across the tetrahedral prisms. The resulting parametrization makes $A=\beta \times [0,1]$, where $\beta$ is its circle or proper-arc frontier on $D_0$. The disk caps of $\beta$ give the asserted subproducts and their boundary conditions. Every fiber stays in its original simplex by that construction. This proves the claimed normal parallelism under the actual containment hypothesis, without any irreducibility assumption on the ambient manifold.