S^{3}$ be a standard Möbius band whose boundary is an unknot $U$. Let $D^{+}$ and $D^{-}$ be standard disks in $B^{4}_{+}$ and $B^{4}_{-}$ whose boundaries are both $U$. These two disks together with $M$ make a foam, $$\Psi = D^{+} \cup D^{-} \cup M.$$ If the half-twist in $M$ has the appropriate sign, the self-intersection of the surface real projective plane $R = D^{+} \cup
M$ in $S^{4}$ will be $-2$, while $D^{-}\cup M$ has self-intersection $+2$. We can realize $(S^{4},\Psi)$, if we wish, as the $V$-quotient of $S^{2}\times S^{2}$, where one generator of $V$ interchanges the two factors and another generator acts as a reflection on each $S^{2}$ (so that the fixed set of the second generator is $S^{1}\times S^{2}$). Let $f_{\pm}$ be interior points in the faces $D^{\pm}$ of $\Psi$.

\[prop:sum-with-double-Mobius\] Let $(X,\Sigma)$ be a foam cobordism with strong marking data $\nu$. Let $f$ be a point in the interior of a face of $\Sigma$. Let $\Psi$ be the foam just described, and let $g \in
    \Psi$ be an interior point of one of the faces of $\Psi$. Let a new foam $\tilde\Sigma$ be constructed from $\Sigma$ as the connected sum $$(X,\Sigma) \#_{f, g} (S^{4},\Psi).$$ Then the new 