# Proof of completeness for shuffles

Proposition (Completeness for shuffles). If [[s1]] [[s2]] then s1 = s2.

Proof. First show by induction on s that:
=
Then show by induction on s that:
=
Then show by induction on s that we can find a shuffle t such that:
=
(The previous result is needed in the case of composition).

Finally, iteratively use the first and last properties to show that if [[s1]] [[s2]] then s1 = s2.

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