Let x0 of type ι be given.
Let x1 of type ο be given.
Assume H1: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x1.
Apply unknownprop_19c5bea28ef119e30d80f4e7c578df826b34bc3d0b15978a12c7c1b896ec3046 with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying L2.
Let x2 of type ι be given.
Assume H3: x2 ∈ x0.
Let x3 of type ι be given.
Assume H4: x3 ∈ x0.
Let x4 of type ι be given.
Assume H5: x4 ∈ x0.
Let x5 of type ι be given.
Assume H6: x5 ∈ x0.
Assume H7: x2 = x3 ⟶ ∀ x6 : ο . x6.
Assume H8: x2 = x4 ⟶ ∀ x6 : ο . x6.
Assume H9: x2 = x5 ⟶ ∀ x6 : ο . x6.
Assume H10: x3 = x4 ⟶ ∀ x6 : ο . x6.
Assume H11: x3 = x5 ⟶ ∀ x6 : ο . x6.
Assume H12: x4 = x5 ⟶ ∀ x6 : ο . x6.
Apply xm with
∀ x6 : ο . (∀ x7 . x7 ∈ x0 ⟶ (x2 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x6) ⟶ x6,
x1 leaving 2 subgoals.
Assume H13: ∀ x6 : ο . (∀ x7 . x7 ∈ x0 ⟶ (x2 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x6) ⟶ x6.
Apply H13 with
x1.
Let x6 of type ι be given.
Assume H14: x6 ∈ x0.
Assume H15: x2 = x6 ⟶ ∀ x7 : ο . x7.
Assume H16: x3 = x6 ⟶ ∀ x7 : ο . x7.
Assume H17: x4 = x6 ⟶ ∀ x7 : ο . x7.
Assume H18: x5 = x6 ⟶ ∀ x7 : ο . x7.
Apply H1 with
x2,
x3,
x4,
x5,
x6 leaving 15 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H14.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H15.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H16.
The subproof is completed by applying H12.
The subproof is completed by applying H17.
The subproof is completed by applying H18.
Assume H13:
not (∀ x6 : ο . (∀ x7 . x7 ∈ x0 ⟶ (x2 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x7 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x6) ⟶ x6).
Let x6 of type ο be given.
Assume H16:
∀ x7 : ι → ι . inj x0 u4 x7 ⟶ x6.
Apply H16 with
inv u4 (λ x7 . ap (lam 4 ...) ...).
Apply FalseE with
x1.
Apply unknownprop_8a6bdce060c93f04626730b6e01b099cc0487102a697e253c81b39b9a082262d with
u4 leaving 2 subgoals.
The subproof is completed by applying nat_4.
Apply atleastp_tra with
u5,
x0,
u4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L14.