Let x0 of type ι be given.
Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x1 . ∀ x2 . In x2 (mul_nat x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . In x4 x0 ⟶ ∀ x5 . In x5 x1 ⟶ x2 = add_nat x4 (mul_nat x5 x0) ⟶ (∀ x6 . In x6 x0 ⟶ ∀ x7 . In x7 x1 ⟶ x2 = add_nat x6 (mul_nat x7 x0) ⟶ and (x6 = x4) (x7 = x5)) ⟶ x3) ⟶ x3 leaving 2 subgoals.
Let x1 of type ι be given.
Apply unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with
x0,
λ x2 x3 . In x1 x3 ⟶ ∀ x4 : ο . (∀ x5 . In x5 x0 ⟶ ∀ x6 . In x6 0 ⟶ x1 = add_nat x5 (mul_nat x6 x0) ⟶ (∀ x7 . In x7 x0 ⟶ ∀ x8 . In x8 0 ⟶ x1 = add_nat x7 (mul_nat x8 x0) ⟶ and (x7 = x5) (x8 = x6)) ⟶ x4) ⟶ x4.
Apply FalseE with
∀ x2 : ο . (∀ x3 . In x3 x0 ⟶ ∀ x4 . In x4 0 ⟶ x1 = add_nat x3 (mul_nat x4 x0) ⟶ (∀ x5 . In x5 x0 ⟶ ∀ x6 . In x6 0 ⟶ x1 = add_nat x5 (mul_nat x6 x0) ⟶ and (x5 = x3) (x6 = x4)) ⟶ x2) ⟶ x2.
Apply unknownprop_1cc88f7e87aaf8c5cee24b4a69ff535a81e7855c45a9fd971eec05ee4cc28f9c with
x1.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
x0,
x1,
λ x3 x4 . In x2 x4 ⟶ ∀ x5 : ο . (∀ x6 . In x6 x0 ⟶ ∀ x7 . In x7 (ordsucc x1) ⟶ x2 = add_nat x6 (mul_nat x7 x0) ⟶ (∀ x8 . In x8 x0 ⟶ ∀ x9 . In x9 (ordsucc x1) ⟶ x2 = add_nat x8 (mul_nat x9 x0) ⟶ and (x8 = x6) (x9 = x7)) ⟶ x5) ⟶ x5 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ο be given.
Assume H5:
∀ x4 . ... ⟶ ∀ x5 . ... ⟶ ... ⟶ (∀ x6 . ... ⟶ ∀ x7 . ... ⟶ x2 = add_nat x6 (mul_nat ... ...) ⟶ and (x6 = x4) (x7 = x5)) ⟶ x3.