Let x0 of type ι be given.
Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x1 . ∀ x2 . In x2 (add_nat x0 x1) ⟶ ∀ x3 : ο . (In x2 x0 ⟶ x3) ⟶ (∀ x4 . In x4 x1 ⟶ x2 = add_nat x0 x4 ⟶ x3) ⟶ x3 leaving 2 subgoals.
Let x1 of type ι be given.
Apply unknownprop_bad5adbbba30ab6e9c584ed350d824b3c3bff74e61c0a5380ac75f32855c37ee with
x0,
λ x2 x3 . In x1 x3 ⟶ ∀ x4 : ο . (In x1 x0 ⟶ x4) ⟶ (∀ x5 . In x5 0 ⟶ x1 = add_nat x0 x5 ⟶ x4) ⟶ x4.
Let x2 of type ο be given.
Assume H1:
In x1 x0 ⟶ x2.
Assume H2:
∀ x3 . In x3 0 ⟶ x1 = add_nat x0 x3 ⟶ x2.
Apply H1.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Assume H1:
∀ x2 . In x2 (add_nat x0 x1) ⟶ ∀ x3 : ο . (In x2 x0 ⟶ x3) ⟶ (∀ x4 . In x4 x1 ⟶ x2 = add_nat x0 x4 ⟶ x3) ⟶ x3.
Let x2 of type ι be given.
Apply unknownprop_bfc870f6d786cc78805c5bf0f9864161d18f532f6daf7daf1d02f4a58dac06f9 with
x0,
x1,
λ x3 x4 . In x2 x4 ⟶ ∀ x5 : ο . (In x2 x0 ⟶ x5) ⟶ (∀ x6 . In x6 (ordsucc x1) ⟶ x2 = add_nat x0 x6 ⟶ x5) ⟶ x5 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ο be given.
Assume H3:
In x2 x0 ⟶ x3.
Apply unknownprop_84fe37a922385756a4e0826a593defb788cadbe4bdc9a7fe6b519ea49f509df5 with
add_nat x0 x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H2.
Apply H1 with
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H3.
Let x4 of type ι be given.
Apply H4 with
x4 leaving 2 subgoals.
Apply unknownprop_9d1f2833af10907d78259d2045ff2d1e1026643f459cca4199c4ae7f89385ba4 with
x1,
x4.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Apply H4 with
x1 leaving 2 subgoals.
The subproof is completed by applying unknownprop_4b3850b342b3607d712ced4e4c9fa37dbdc70692760e3dc82f8fd86e9b26a6b5 with x1.
The subproof is completed by applying H5.