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Let x0 of type ι → ο be given.
Apply unknownprop_1db1571afe8c01990252b7801041a0001ba1fedff9d78947d027d61a0ff0ae7f with x0, λ x1 . ap x1 0, UnaryPredHom leaving 3 subgoals.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x0 x1.
Assume H2: x0 x2.
Apply H0 with x1, λ x4 . UnaryPredHom x4 x2 x3 ⟶ x3 ∈ setexp (ap x2 0) (ap x4 0) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Let x5 of type ι → ο be given.
Apply H0 with x2, λ x6 . UnaryPredHom (pack_p x4 x5) x6 x3 ⟶ x3 ∈ setexp (ap x6 0) (ap (pack_p x4 x5) 0) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x6 of type ι be given.
Let x7 of type ι → ο be given.
Apply unknownprop_63c01b8f599732ba7bc3b48c28c0f10755230e5cc9f0717c7895602d2eaf01d3 with x4, x6, x5, x7, x3, λ x8 x9 : ο . x9 ⟶ x3 ∈ setexp (ap (pack_p x6 x7) 0) (ap (pack_p x4 x5) 0).
Assume H3: and (x3 ∈ setexp x6 x4) (∀ x8 . x8 ∈ x4 ⟶ x5 x8 ⟶ x7 (ap x3 x8)). Apply H3 with x3 ∈ setexp (ap (pack_p x6 x7) 0) (ap (pack_p x4 x5) 0).
Assume H5: ∀ x8 . x8 ∈ x4 ⟶ x5 x8 ⟶ x7 (ap x3 x8). Apply pack_p_0_eq2 with x6, x7, λ x8 x9 . x3 ∈ setexp x8 (ap (pack_p x4 x5) 0).
Apply pack_p_0_eq2 with x4, x5, λ x8 x9 . x3 ∈ setexp x6 x8.
The subproof is completed by applying H4.
Let x1 of type ι be given.
Assume H1: x0 x1.
Apply H0 with x1, λ x2 . UnaryPredHom x2 x2 (lam_id (ap x2 0)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Let x3 of type ι → ο be given.
Apply pack_p_0_eq2 with x2, x3, λ x4 x5 . UnaryPredHom (pack_p x2 x3) (pack_p x2 x3) (lam_id x4).
Apply unknownprop_63c01b8f599732ba7bc3b48c28c0f10755230e5cc9f0717c7895602d2eaf01d3 with x2, x2, x3, x3, lam_id x2, λ x4 x5 : ο . x5.
Apply andI with lam_id x2 ∈ setexp x2 x2, ∀ x4 . x4 ∈ x2 ⟶ x3 x4 ⟶ x3 (ap (lam_id x2) x4) leaving 2 subgoals.
The subproof is completed by applying lam_id_exp_In with x2.
Let x4 of type ι be given.
Assume H2: x4 ∈ x2.
Assume H3: x3 x4.
Apply beta with x2, λ x5 . x5, x4, λ x5 x6 . x3 x6 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H1: x0 x1.
Assume H2: x0 x2.
Assume H3: x0 x3.
Apply H0 with x1, λ x6 . UnaryPredHom x6 x2 x4 ⟶ UnaryPredHom x2 x3 x5 ⟶ UnaryPredHom x6 x3 (lam_comp (ap x6 0) x5 x4) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x6 of type ι be given.
Let x7 of type ι → ο be given.
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