Small 2-connected subgraphs localise in high-girth pieces

By the occurrence convention, choose an isomorphism $f:B\to H$. The map $\{u,v\}\mapsto \{f(u),f(v)\}$ is a bijection from $E(B)$ to $E(H)$: adjacency preservation makes it well-defined, injectivity of $f$ makes it injective on unordered pairs, and adjacency reflection and surjectivity of $f$ make it surjective. Thus $H$ has $m$ edges, and the vertex bijection gives it at least three vertices. Applying $f$ to a path preserves distinctness of its vertices and adjacency of consecutive vertices. Paths in the connected graph $B$ therefore show that $H$ is connected. For any vertex $h=f(v)$ of $H$, paths between any two other preimages in $B$ after deletion of $v$ map to paths between the corresponding vertices of $H$ avoiding $h$. Thus $H$ remains connected after deletion of any one vertex, so $H$ is 2-connected.

For each of the finitely many edges of $H$, choose one hyperedge whose piece supplies that edge; such a hyperedge exists by the union defining $R$. Denote the chosen hyperedge for an edge $a$ by $c(a)$, and let $\mathcal{F}=\{c(a):a\in E(H)\}$. Choose a listing without repetition $a_0,\dots ,a_{m-1}$ of $E(H)$, which exists by its cardinality $m$.

For each $e\in \mathcal{F}$, the set of indices $i$ with $0\le i<m$ and $c(a_i)=e$ is a nonempty subset of $\mathbb{N}$. Let $i(e)$ be its least element, supplied by F7, and set $D=\{a_{i(e)}:e\in \mathcal{F}\}\subseteq E(H)$. The map $e\mapsto a_{i(e)}$ is surjective onto $D$ by its definition. It is injective because $a_{i(e)}=a_{i(e^{\mathrm{\prime }})}$ implies $e=c(a_{i(e)})=c(a_{i(e^{\mathrm{\prime }})})=e^{\mathrm{\prime }}$. Thus its inverse, which sends $a\in D$ to $c(a)$, is a bijection from $D$ to $\mathcal{F}$. The finite-subset bound makes $D$ finite with $\mathrm{\mid }D\mathrm{\mid }\le \mathrm{\mid }E(H)\mathrm{\mid }=m$. Cardinality transport along this bijection makes $\mathcal{F}$ finite with $\mathrm{\mid }\mathcal{F}\mathrm{\mid }=\mathrm{\mid }D\mathrm{\mid }\le m<q$.

Form the restricted hypergraph $(V,\mathcal{F})$ and realise it using the same pieces indexed by $\mathcal{F}$. Its realisation has vertex set $V$ and contains every edge of $H$ by the choice of $c$, so $H$ is its ordinary subgraph.

By the support-forest result, the incidence graph on the selected hyperedge nodes and their incident vertex nodes is a forest. The incidence graph of $(V,\mathcal{F})$ adds only isolated vertex nodes, which cannot belong to a cycle, and is therefore also a forest. The forest-realisation result now puts the 2-connected subgraph $H$ inside one selected piece. That piece is also a piece of $R$.

An occurrence of $K_2$ consists of one edge and its endpoints, since its isomorphism with $K_2$ is a vertex bijection preserving and reflecting adjacency. The edge belongs to some piece by the defining union, and that piece contains its endpoints; hence the occurrence is contained in that piece. If $B$ is as in the final assertion and an occurrence existed in $R$, step 5.1 or this argument would put it in a piece, contrary to the assumed exclusion.