proposition: Lemma comultiplication_comonoid_4times (m : commutative_comonoid M) : δ_{m} · m ⊗^{M}_{l} δ_{ m} · m ⊗^{M}_{l} (δ_{ m} ⊗^{M}_{r} m) = δ_{m} · δ_{m} ⊗^{M} δ_{ m} · mon_lassociator _ _ (_ ⊗ m) · m ⊗^{M}_{l} mon_rassociator _ _ _.
proof: Proof. etrans. 2: { apply maponpaths_2. exact (! commutative_symmetric_braiding_using_lwhisker _). } rewrite ! assoc'. apply maponpaths. rewrite <- (bifunctor_leftcomp M). etrans. { apply pathsinv0, (bifunctor_leftcomp M). } etrans. 2: { apply (bifunctor_leftcomp M). } apply maponpaths. refine (comonoid_laws_assoc' M m @ _). apply assoc'. Qed.
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