Let x0 of type ι → ο be given.

Let x1 of type ι → ι → ι → ο be given.

Let x2 of type ι → ι be given.

Let x3 of type ι → ι → ι → ι → ι → ι be given.

Assume H0: MetaFunctor_prop1 x0 x1 x2 x3.

Assume H1: MetaFunctor_prop2 x0 x1 x2 x3.

Assume H2: ∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x4 x5 x6 (x2 x4) = x6.

Assume H3: ∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x5 x5 (x2 x5) x6 = x6.

Assume H4: ∀ x4 x5 x6 x7 x8 x9 x10 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x1 x4 x5 x8 ⟶ x1 x5 x6 x9 ⟶ x1 x6 x7 x10 ⟶ x3 x4 x5 x7 (x3 x5 x6 x7 x10 x9) x8 = x3 x4 x6 x7 x10 (x3 x4 x5 x6 x9 x8).

Apply and3I with and (MetaFunctor_prop1 x0 x1 x2 x3) (MetaFunctor_prop2 x0 x1 x2 x3), and (MetaCat_IdR_p x0 x1 x2 x3) (MetaCat_IdL_p x0 x1 x2 x3), MetaCat_Comp_assoc_p x0 x1 x2 x3 leaving 3 subgoals.

Apply andI with MetaFunctor_prop1 x0 x1 x2 x3, MetaFunctor_prop2 x0 x1 x2 x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Apply andI with MetaCat_IdR_p x0 x1 x2 x3, MetaCat_IdL_p x0 x1 x2 x3 leaving 2 subgoals.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

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Proofgold Proof