Remark 5.4.4.1.2 (00FB) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

Unwinding the notation for elements, we have

\begin{aligned} [(x, [(y, z)])] & \overset{def}{=} [(x, y ⊳ z)] \\ \overset{def}{=} x ⊳ (y ⊳ z) \end{aligned}

and

\begin{aligned} [([(x, y)], z)] & \overset{def}{=} [(x ⊳ y, z)] \\ \overset{def}{=} (x ⊳ y) ⊳ z . \end{aligned}

So, in other words, $α_{X, Y, Z}^{{Sets}_{*}, ⊳}$ acts on elements via

α_{X, Y, Z}^{{Sets}_{*}, ⊳} (x ⊳ (y ⊳ z)) \overset{def}{=} (x ⊳ y) ⊳ z

for each $x ⊳ (y ⊳ z) \in X ⊳ (Y ⊳ Z)$ .

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