Section ChainCompose.
Variables (T1 T2 T3 : poset) (f1 : T2 -> T1) (f2 : T3 -> T2).
Variables (s : chain T3) (M1 : monotone f1) (M2 : monotone f2).

Lemma comp_chainE :
[f1 ^^ [f2 ^^ s by M2] by M1] = [f1 \o f2 ^^ s by comp_mono M1 M2].