= Fano completion through three matchings
{c}
If $xyz$ is a <hyperedge> and the links of $x,y,z$ on four other vertices contain the three perfect matchings of a $K_4$, then those seven vertices contain a <Fano plane>. There is a second completion: if the link of $v$ contains three disjoint pairs, and the other six vertices contain all transversal triples through those pairs, take four triples of even parity and the three triples containing $v$. This again gives a Fano plane. These finite templates convert forbidden Fano copies into missing cross-part triples.
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