Let x0 of type ι → ι → ο be given.
Assume H0: x0 4 5.
Assume H1: ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1.
Apply xm with
x0 0 4,
or (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3))) (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)))) leaving 2 subgoals.
Assume H3: x0 0 4.
Apply unknownprop_aaed795e8d5c2653e9f652bec86f5ef354e81828f5a08710e4e31ae5a49671af with
x0 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply xm with
x0 1 4,
or (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3))) (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)))) leaving 2 subgoals.
Assume H4: x0 1 4.
Apply L5 with
or (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3))) (∃ x1 . and (x1 ⊆ 6) (and (equip 3 x1) (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3)))).
Let x1 of type ι → ι be given.
Assume H6:
(λ x2 : ι → ι . and (and (x2 0 = 1) (x2 1 = 0)) (∀ x3 . (x3 = 0 ⟶ ∀ x4 : ο . x4) ⟶ (x3 = 1 ⟶ ∀ x4 : ο . x4) ⟶ x2 x3 = x3)) x1.
Apply H6 with
or (∃ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4))) (∃ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4)))).