= Power-set closure of better-quasi-orderings
{title2=$A\le_H B\iff\forall a\in A\ \exists b\in B\ (a\le b)$}
If $Q$ is a <better-quasi-ordering>, its full <power set> is better-quasi-ordered by the <Hoare domination preorder>. A bad continuous array of subsets would allow a choice of $q_X\in A_X$ not below any member of $A_{X^-}$, where $X^-$ deletes the least element. Choose this witness as a fixed function of the two locally constant subsets. Then $X\mapsto q_X$ is a bad <continuous array in better-quasi-order theory> into $Q$, a contradiction. Empty source subsets cannot occur in a bad array because they precede every subset.
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