Proposition/definition 36: Given a binary predicate \sim , we may attach the superscript \hspace{0.1mm}^\textup{op} to the predicate expression to get a new binary predicate (x \sim^\textup{op} y) \hspace{3mm} \textup{defined by} \hspace{3mm} (y \sim x). We call this the reverse (binary) predicate of \sim . It follows that (x \hspace{1mm} {\sim^\textup{op}}^\textup{op} y) \hspace{3mm} \textup{is the same as} \hspace{3mm} (x \sim y). If the predicate expression is single symbol which is not symmetric across its central vertical axis, we may horizontally mirror the symbol to write (x \backsim y) \hspace{3mm} \textup{in place of} \hspace{3mm} (y \sim^\textup{op} x), noting that mirroring it again returns the original predicate symbol.

Proof: (x \hspace{1mm} {\sim^\textup{op}}^\textup{op} \hspace{1mm} y) is by definition given by (y \sim^\textup{op} x) , which is once again by definition given by (x \sim y) . \square