Let x0 of type ι be given.
Apply unknownprop_5273a64cd2d40c560fbf5871c7601206337acc93fe8f7292cc8c65a171682421 with
λ x1 x2 : ι → ο . x2 x0 ⟶ x2 (ordsucc x0).
Apply andE with
TransSet x0,
∀ x1 . In x1 x0 ⟶ TransSet x1,
(λ x1 . and (TransSet x1) (∀ x2 . In x2 x1 ⟶ TransSet x2)) (ordsucc x0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
TransSet (ordsucc x0),
∀ x1 . In x1 (ordsucc x0) ⟶ TransSet x1 leaving 2 subgoals.
Apply unknownprop_c737afbfc89fda2c77b5fd830c84180fdf657c880ee26d42c23167ad83f331dd with
x0.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply unknownprop_84fe37a922385756a4e0826a593defb788cadbe4bdc9a7fe6b519ea49f509df5 with
x0,
x1,
TransSet x1 leaving 3 subgoals.
The subproof is completed by applying H3.
Apply H2 with
x1.
The subproof is completed by applying H4.
Assume H4: x1 = x0.
Apply H4 with
λ x2 x3 . TransSet x3.
The subproof is completed by applying H1.