### Proof

Let WLSPA hold over the set of orderings \(\mathscr {D}|A\) of triple \(A = \{x,y,z\}\).

(i) If \(\mathscr {D}|A\) does not contain a weak Latin Square then VR(1) is satisfied; as a set of orderings of a triple violates VR(1) iff the set contains a weak Latin Square.

(ii) Suppose \(\mathscr {D}|A\) contains one of the two weak Latin Squares, say *LS*(*xyzx*), and does not contain the other one. This implies that:

\(\mathscr {D}|A\) does not contain the ordering *xIyIz*. (P3.6-1)

As \(\mathscr {D}|A\) does not contain *WLS*(*xzyx*) we have: \(\sim [(\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A)(xR^{s}zR^{s}y \wedge zR^{t}yR^{t}x \wedge yR^{u}xR^{u}z)]\). Without any loss of generality assume that \(\mathscr {D}|A\) does not contain any ordering *R* such that *xRzRy*. This implies that we must have:

\((\forall R \in \mathscr {D}|A)[(xRz \rightarrow yPz) \wedge (zRy \rightarrow zPx)]\). (P3.6-2)

Consequently, we have:

\((\forall R \in \mathscr {D}|A)[(xRyRz \rightarrow xRyPz) \wedge (zRxRy \rightarrow zPxRy)]\). (P3.6-3)

If no \(R {\in } \mathscr {D}|A \cap T[LS(xyzx)]\) is strong then (P3.6-3) implies that \((\forall R {\in } \mathscr {D}|A)(yRx)\), and therefore, in view of (P3.6-1), TP is satisfied.

Next consider the case when some ordering belonging to \(\mathscr {D}|A \cap T[LS(xyzx)]\) is strong. This strong ordering cannot be *xPyPz* in view of *zPxRy* belonging to \(\mathscr {D}|A\), otherwise WLSPA would be violated. This strong ordering cannot be *zPxPy* either in view of *xRyPz* belonging to \(\mathscr {D}|A\), otherwise WLSPA would be violated. Thus, if an \(R \in \mathscr {D}|A \cap T[LS(xyzx)]\) is strong, it has to be *yPzPx*. Once again, it follows that: \((\forall R \in \mathscr {D}|A)(yRx)\), and therefore, in view of (1), TP is satisfied.

(iii) Finally consider the case when \(\mathscr {D}|A\) contains both the weak Latin Squares. If no ordering belonging to \(\mathscr {D}|A\) is strong, then DP holds, and consequently, ER is satisfied.

Suppose \(\mathscr {D}|A\) contains a strong ordering, say, *xPyPz*.

WLSPA holds over \(\mathscr {D}|A\) \(\wedge \) \(\mathscr {D}|A\) contains both the weak Latin Squares \(\wedge \) \(\mathscr {D}|A\) contains *xPyPz* \(\rightarrow \) \((\forall R \in \mathscr {D}|A)[(yRzRx \rightarrow yRzIx) \wedge (zRxRy \rightarrow zIxRy)]\) (P3.6-4)

(P3.6-4) implies that \(\mathscr {D}|A\) does not contain any of *yPzPx*, *yIzPx*, *zPxPy*, *zPxIy* (P3.6-5)

If *xPyPz* is the only strong ordering in \(\mathscr {D}|A\) then (P3.6-5) implies that \((\forall R \in \mathscr {D}|A)(xRz)\). Therefore, EP holds, and consequently, ER is satisfied.

Next suppose that, in addition to *xPyPz*, \(\mathscr {D}|A\) contains another strong ordering. In view of (P3.6-5), it follows that this strong ordering must belong to *WLS*(*xzyx*).

First consider the case when this other strong ordering contained in \(\mathscr {D}|A\) is *xPzPy*.

WLSPA holds over \(\mathscr {D}|A\) \(\wedge \) \(\mathscr {D}|A\) contains both the weak Latin Squares \(\wedge \) \(\mathscr {D}|A\) contains *xPzPy* \(\rightarrow \) \((\forall R \in \mathscr {D}|A)[(yRxRz \rightarrow yIxRz) \wedge (zRyRx \rightarrow zRyIx)]\) (P3.6-6)

(P3.6-6) implies that \(\mathscr {D}|A\) does not contain any of *yPxPz*, *yPxIz*, *zPyPx*, *zIyPx* (P3.6-7)

(P3.6-5) and (P3.6-7) imply that \(\mathscr {D}|A \subseteq \{xPyPz, xPzPy, xPyIz, xIyPz, xIzPy, xIyIz\}\). Therefore, EP holds, and consequently, ER is satisfied.

Next consider the case when this other strong ordering contained in \(\mathscr {D}|A\) is *yPxPz*.

WLSPA holds over \(\mathscr {D}|A\) \(\wedge \) \(\mathscr {D}|A\) contains both the weak Latin Squares \(\wedge \) \(\mathscr {D}|A\) contains *yPxPz* \(\rightarrow \) \((\forall R \in \mathscr {D}|A)[(xRzRy \rightarrow xRzIy) \wedge (zRyRx \rightarrow zIyRx)]\) (P3.6-8)

(P3.6-8) implies that \(\mathscr {D}|A\) does not contain any of *xPzPy*, *xIzPy*, *zPyPx*, *zPyIx* (P3.6-9)

(P3.6-5) and (P3.6-9) imply that \(\mathscr {D}|A \subseteq \{xPyPz, yPxPz, xPyIz, xIyPz, yPxIz, xIyIz\}\). Therefore, EP holds, and consequently, ER is satisfied.

Finally consider the case when this other strong ordering contained in \(\mathscr {D}|A\) is *zPyPx*.

WLSPA holds over \(\mathscr {D}|A\) \(\wedge \) \(\mathscr {D}|A\) contains both the weak Latin Squares \(\wedge \) \(\mathscr {D}|A\) contains *zPyPx* \(\rightarrow \) \((\forall R \in \mathscr {D}|A)[(xRzRy \rightarrow xIzRy) \wedge (yRxRz \rightarrow yRxIz)]\) (P3.6-10)

(P3.6-10) implies that \(\mathscr {D}|A\) does not contain any of *xPzPy*, *xPzIy*, *yPxPz*, *yIxPz* (P3.6-11)

(P3.6-5) and (P3.6-11) imply that \(\mathscr {D}|A {\subseteq } \{xPyPz, zPyPx, yPzIx, zIxPy, xIyIz\}\). Therefore, AP holds, and consequently, ER is satisfied.

This completes the proof of the assertion that satisfaction of WLSPA implies satisfaction of at least one of the three conditions of value restriction (1), taboo preferences and extremal restriction.

Let \(\mathscr {D}|A\) violate WLSPA. Without any loss of generality, assume that \(\mathscr {D}|A\) contains *WLS*(*xyzx*) involving a strong ordering, say *xPyPz*, and also contains an ordering belonging to *LS*(*xyzx*) with *zPx*. That is to say, \((\exists R^{s},R^{t},R^{u} \in \mathscr {D}|A)[(xP^{s}yP^{s}z \wedge yR^{t}zP^{t}x \wedge zR^{u}xR^{u}y) \vee (xP^{s}yP^{s}z \wedge yR^{t}zR^{t}x \wedge zP^{u}xR^{u}y)].\) (P3.6-12)

\(\mathscr {D}|A\) contains \(R^{s},R^{t},R^{u}\) such that \((xP^{s}yP^{s}z \wedge yR^{t}zR^{t}x \wedge zR^{u}xR^{u}y)\) implies that VR(1) is violated. (P3.6-13)

\(xP^{s}yP^{s}z {\in } \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (xPy) \wedge (\exists R \in \mathscr {D}|A) (yPz) \wedge (\exists R \in \mathscr {D}|A) (xPz)\) (P3.6-14)

\(yR^{t}zR^{t}x \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (yPx \vee xIyIz)\) (P3.6-15)

\(zR^{u}xR^{u}y \in \mathscr {D}|A \rightarrow (\exists R \in \mathscr {D}|A) (zPy \vee xIyIz)\) (P3.6-16)

(P3.6-12) \(\rightarrow (\exists R \in \mathscr {D}|A) (zPx)\) (P3.6-17)

\(xIyIz \in \mathscr {D}|A\) implies that TP is violated. (P3.6-18)

From (P3.6-14)-(P3.6-17) it follows that if \(xIyIz \notin \mathscr {D}|A\) then also TP is violated. (P3.6-19)

(P3.6-18) and (P3.6-19) establish that TP is violated. (P3.6-20)

From (P3.6-12) it follows that: \((xP^{s}yP^{s}z, yR^{t}zP^{t}x \in \mathscr {D}|A) \vee (xP^{s}yP^{s}z, zP^{u}xR^{u}y \in \mathscr {D}|A)\) (P3.6-21)

(P3.6-21) implies that ER is violated. (P3.6-22)

(P3.6-13), (P3.6-20), and (P3.6-22) establish the assertion that if a set of orderings of a triple satisfies at least one of the three conditions of value restriction (1), taboo preferences and extremal restriction, then it satisfies WLSPA. \(\square \)