Both completions are the Fano plane

Setting. Throughout, the point set is $\mathbb{Z}/7 = \{0,1,\dots,6\}$ (the type of natural numbers below $7$, with addition taken modulo $7$). A Steiner triple system on $n$ points (in the sense used here) is a triple consisting of

• a finite collection $\mathcal{L}$ of subsets of $\{0,\dots,n-1\}$ (the lines),
• the property that every line $l \in \mathcal{L}$ has exactly $3$ elements,
• the property that for all points $x \neq y$ there is exactly one line $l \in \mathcal{L}$ with $x \in l$ and $y \in l$.

The theorem takes an arbitrary such system $S$ on $n = 7$ points, with line collection $\mathcal{L}_S$ (the two axioms are part of the data of $S$).

Hypothesis. $\mathcal{L}_S$ is literally equal (as a set of $3$-element subsets of $\mathbb{Z}/7$) to one of the following two collections:

That is, $\mathcal{L}_S = \mathcal{A}$ or $\mathcal{L}_S = \mathcal{B}$ (an inclusive "or"). Each of $\mathcal{A}$ and $\mathcal{B}$ consists of seven distinct $3$-element sets, and in each every pair of distinct points of $\mathbb{Z}/7$ lies in exactly one of the seven sets, so both alternatives are compatible with the Steiner-system axioms; the hypothesis is not vacuous.

Reference system. The fixed "Fano" system $F$ on $\mathbb{Z}/7$ has as lines the sets $\{i,\ i+1,\ i+3\}$ (addition mod $7$) for $i = 0,\dots,6$, i.e.

Conclusion. $S$ "is Fano", which by definition means: there exists a bijection $e : \mathbb{Z}/7 \to \mathbb{Z}/7$ (a permutation of the seven points) such that

where $e(l) = \{e(x) : x \in l\}$ is the pointwise image of the line $l$. In words: under either hypothesis on the lines of $S$, some relabelling of the points carries the set of lines of $S$ exactly onto the set of lines of the reference system $F$ (equality of collections of lines, so every line of $S$ maps to a line of $F$ and every line of $F$ is the image of some line of $S$). Only existence of such a permutation is asserted; no uniqueness is claimed.