$\Jsharp$, in which case we deduce the version in the introduction, Theorem \[thm:intro-three-bar\].

There is a similar generalization of Theorem \[thm:intro-one-bar\] in the setting of foams in a $3$-manifold $Y$ with strong marking, whose statement is easily formulated. It will turn out that there is an argument that allows Theorem \[thm:intro-one-bar\] to be deduced from Theorem \[thm:intro-three-bar\] (or, in the more general form, from Theorem \[thm:exact-triangle\]). We will therefore focus on Theorem \[thm:exact-triangle\] to begin with.

Calculations for some connected sums {#sec:connect-sums}
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The quotient of $-\CP^{2}$ by the action of complex conjugation, $[z_{1}, z_{2}, z_{3}]\mapsto [\bar z_{1}, \bar z_{2},\bar z_{3}]$, is an orbifold $S^{4}$ containing as branch locus the image $R\subset
S^{4}$ of $\RP^{2}\subset -\CP^{2}$. If $L$ is a complex line in $-\CP^{2}$ defined by real linear equation in the homogeneous coordinates, then the image of $L$ in the quotient is a disk $D$ whose boundary is a real projective line in the branch locus $R$. Given $n$ such lines in $-\CP^{2}$, say $$L_{1} ,\dots, L_{n},$$ we obtain $n$ disks $D_{1} ,\dots, D_{n}$ in $S^{4}$ whose interiors are disjoint and whose boundaries meet in pairs at points of $R$. The union $$\label{eq:Psi-foams}
         \Psi_{n} = R \cup D_{1} \cup \cdots 