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.
Let x4 of type ι → ο be given.
Let x5 of type ι → ι → ι → ο be given.
Let x6 of type ι → ι be given.
Let x7 of type ι → ι → ι → ι → ι → ι be given.
Let x8 of type ι → ι be given.
Let x9 of type ι → ι → ι → ι be given.
Let x10 of type ι → ι be given.
Let x11 of type ι → ι → ι → ι be given.
Let x12 of type ι → ι be given.
Let x13 of type ι → ι be given.
Apply unknownprop_9a5dd92d37ccfa65696c11e832d98097811bf4001ca7eb00f4f9586fc6e6bb6b with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
∀ x14 . ∀ x15 : ι → ι . MetaCat_initial_p x0 x1 x2 x3 x14 x15 ⟶ ∃ x16 : ι → ι . MetaCat_initial_p x4 x5 x6 x7 (x8 x14) x16 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2:
MetaFunctor x4 x5 x6 x7 x0 x1 x2 x3 x10 x11.
Assume H3:
MetaNatTrans x0 x1 x2 x3 x0 x1 x2 x3 (λ x14 . x14) (λ x14 x15 x16 . x16) (λ x14 . x10 (x8 x14)) (λ x14 x15 x16 . x11 (x8 x14) (x8 x15) (x9 x14 x15 x16)) x12.
Assume H4:
MetaNatTrans x4 x5 x6 x7 x4 x5 x6 x7 (λ x14 . x8 (x10 x14)) (λ x14 x15 x16 . x9 (x10 x14) (x10 x15) (x11 x14 x15 x16)) (λ x14 . x14) (λ x14 x15 x16 . x16) x13.
Assume H5:
MetaAdjunction x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Apply unknownprop_67ed42aa94f161ee21a2e65a66bf8b96dc66d4484eee9eeb2abcb112d8b49161 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
∀ x14 . ∀ x15 : ι → ι . MetaCat_initial_p x0 x1 x2 x3 x14 x15 ⟶ ∃ x16 : ι → ι . MetaCat_initial_p x4 x5 x6 x7 (x8 x14) x16 leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6: ∀ x14 . x0 x14 ⟶ x7 (x8 x14) (x8 (x10 (x8 x14))) (x8 x14) (x13 (x8 x14)) (x9 x14 (x10 (x8 x14)) (x12 x14)) = x6 (x8 x14).
Assume H7: ∀ x14 . x4 x14 ⟶ x3 (x10 x14) (x10 (x8 (x10 x14))) (x10 x14) (x11 (x8 (x10 x14)) x14 (x13 x14)) (x12 (x10 x14)) = x2 (x10 x14).
Apply unknownprop_b9f4ecece16a3f4b44463b508cc3b9f5d1731684163a4bbdbf54ad9580b00fef with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
∀ x14 . ∀ x15 : ι → ι . MetaCat_initial_p x0 x1 x2 x3 x14 x15 ⟶ ∃ x16 : ι → ι . MetaCat_initial_p x4 x5 x6 x7 (x8 x14) x16 leaving 2 subgoals.
The subproof is completed by applying H1.