Let x0 of type ι → ο be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ι → ο be given.
Assume H0:
∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x2 x3 x4 x5 ⟶ x5 ∈ setexp (x1 x4) (x1 x3).
Assume H1:
∀ x3 . x0 x3 ⟶ x2 x3 x3 (lam_id (x1 x3)).
Assume H2:
∀ x3 x4 x5 x6 x7 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 x4 x6 ⟶ x2 x4 x5 x7 ⟶ x2 x3 x5 (lam_comp (x1 x3) x7 x6).
Apply unknownprop_fc5379bc4ad65dc1954d6f65361b9d804f439ab0844013155adf361a615275a6 with
x0,
x2,
λ x3 . lam_id (x1 x3),
λ x3 x4 x5 x6 x7 . lam_comp (x1 x3) x6 x7 leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x2 x3 x4 x5.
Apply lam_comp_id_R with
x1 x3,
x1 x4,
x5.
Apply H0 with
x3,
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x2 x3 x4 x5.
Apply lam_comp_id_L with
x1 x3,
x1 x4,
x5.
Apply H0 with
x3,
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x2 x3 x4 x7.
Assume H8: x2 x4 x5 x8.
Assume H9: x2 x5 x6 x9.
Let x10 of type ι → ι → ο be given.
Apply lam_comp_assoc with
x1 x3,
x1 x4,
x7,
x8,
x9,
λ x11 x12 . x10 x12 x11.
Apply H0 with
x3,
x4,
x7 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H7.