Let x0 of type ι be given.
Apply SNoLtLe_or with
x0,
0,
abs_SNo (minus_SNo x0) = abs_SNo x0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply neg_abs_SNo with
x0,
λ x1 x2 . abs_SNo (minus_SNo x0) = x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNoLtLe with
0,
minus_SNo x0.
Apply minus_SNo_Lt_contra2 with
x0,
0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply minus_SNo_0 with
λ x1 x2 . SNoLt x0 x2.
The subproof is completed by applying H1.
Apply nonneg_abs_SNo with
minus_SNo x0.
The subproof is completed by applying L2.
Apply SNoLtLe_or with
minus_SNo x0,
0,
abs_SNo (minus_SNo x0) = abs_SNo x0 leaving 4 subgoals.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply nonneg_abs_SNo with
x0,
λ x1 x2 . abs_SNo (minus_SNo x0) = x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply neg_abs_SNo with
minus_SNo x0,
λ x1 x2 . x2 = x0 leaving 3 subgoals.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply minus_SNo_invol with
x0.
The subproof is completed by applying H0.
Claim L3: x0 = 0
Apply SNoLe_antisym with
x0,
0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply minus_SNo_0 with
λ x1 x2 . SNoLe x0 x1.
Apply minus_SNo_invol with
x0,
λ x1 x2 . SNoLe x1 (minus_SNo 0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply minus_SNo_Le_contra with
0,
minus_SNo x0 leaving 3 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply L3 with
λ x1 x2 . abs_SNo (minus_SNo x2) = abs_SNo x2.
Claim L4: ∀ x3 : ι → ο . x3 y2 ⟶ x3 y1
Let x3 of type ι → ο be given.
set y4 to be λ x4 . x3
Apply minus_SNo_0 with
λ x5 x6 . y4 (abs_SNo x5) (abs_SNo x6).
The subproof is completed by applying H4.
Let x3 of type ι → ι → ο be given.
Apply L4 with
λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H5: x3 y2 y2.
The subproof is completed by applying H5.