Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

Claim L2: SNo (minus_SNo x0)

Apply SNo_minus_SNo with x0.

The subproof is completed by applying H0.

Claim L3: SNo (minus_SNo x1)

Apply SNo_minus_SNo with x1.

The subproof is completed by applying H1.

Claim L4: SNo (add_SNo x1 (minus_SNo x0))

Apply SNo_add_SNo with x1, minus_SNo x0 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying L2.

set y2 to be abs_SNo (add_SNo x0 (minus_SNo x1))

set y3 to be abs_SNo (add_SNo y2 (minus_SNo x1))

Claim L5: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2

Let x4 of type ι → ο be given.

Assume H5: x4 (abs_SNo (add_SNo y3 (minus_SNo y2))).

set y5 to be abs_SNo (add_SNo y2 (minus_SNo y3))

Claim L6: ∀ x7 : ι → ο . x7 y6 ⟶ x7 y5

Let x7 of type ι → ο be given.

set y8 to be λ x8 . x7

Apply minus_add_SNo_distr with y5, minus_SNo x4, λ x9 x10 . add_SNo x4 (minus_SNo y5) = x10, λ x9 x10 . y8 (abs_SNo x9) (abs_SNo x10) leaving 4 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying L2.

Apply minus_SNo_invol with x4, λ x9 x10 . add_SNo x4 (minus_SNo y5) = add_SNo (minus_SNo y5) x10 leaving 2 subgoals.

The subproof is completed by applying H0.

Apply add_SNo_com with x4, minus_SNo y5 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying L3.

The subproof is completed by applying H6.

set y7 to be λ x7 . y6

Apply L6 with λ x8 . y7 x8 y6 ⟶ y7 y6 x8 leaving 2 subgoals.

Assume H7: y7 y6 y6.

The subproof is completed by applying H7.

Apply abs_SNo_minus with add_SNo y6 (minus_SNo y5), λ x8 . y7 leaving 2 subgoals.

The subproof is completed by applying H5.

The subproof is completed by applying L6.

Let x4 of type ι → ι → ο be given.

Apply L5 with λ x5 . x4 x5 y3 ⟶ x4 y3 x5.

Assume H6: x4 y3 y3.

The subproof is completed by applying H6.

■

Proofgold Proof