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Why Is Pudlak's Relation on the Family of One- or Two-Element Subsets of a Set Transitive?

The following comes from Definition 2 in Pavel Pudlak, "A new proof of the congruence lattice representation theorem," Algebra Universalis 6 (1976), 269-275.

Let $X$ be a set. Let $F$ be a family of functions from $X$ to itself containing the identity map and closed under composition.

Define a binary relation on the family of one- or two-element subsets of $X$ as follows. Let $a,b,c,d\in X$. We will say that $\{a,b\}$ dominates $\{c,d\}$ if there are $n\in\mathbb N_0$, $u_0,\dots,u_n\in X$, and $f_1,\dots,f_n\in F$ such that $u_0=c$, $u_n=d$, and $\{f_i(a),f_i(b)\}=\{u_{i-1},u_i\}$ for $i=1,\dots,n$.

Why is domination transitive?

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