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.
Apply unknownprop_d158dd0ee7cd129998fef286b8e8d5798d9fdd1daf331b8339c3836f8066dce4 with
x0,
x1,
x2,
x3,
∀ x4 x5 : ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p x0 x1 x2 x3 x4 x5 x6 ⟶ ∀ x7 x8 x9 : ι → ι → ι . ∀ x10 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 x1 x2 x3 x7 x8 x9 x10 ⟶ MetaCat_pullback_struct_p x0 x1 x2 x3 (λ x11 x12 x13 x14 x15 . x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (λ x11 x12 x13 x14 x15 . x3 (x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (x7 x11 x12) x11 (x8 x11 x12) (x5 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)))) (λ x11 x12 x13 x14 x15 . x3 (x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (x7 x11 x12) x12 (x9 x11 x12) (x5 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)))) (λ x11 x12 x13 x14 x15 x16 x17 x18 . x6 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)) x16 (x10 x11 x12 x16 x17 x18)) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: ∀ x4 . x0 x4 ⟶ x1 x4 x4 (x2 x4).
Assume H2: ∀ x4 x5 x6 x7 x8 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x1 x4 x5 x7 ⟶ x1 x5 x6 x8 ⟶ x1 x4 x6 (x3 x4 x5 x6 x8 x7).
Assume H3: ∀ x4 x5 x6 . ... ⟶ ... ⟶ ... ⟶ x3 x4 x4 ... ... ... = ....