Let x0 of type ι be given.
Let x1 of type ι be given.
Apply minus_add_SNo_distr with
minus_SNo x0,
x1,
λ x2 x3 . x3 = add_SNo x0 (minus_SNo x1) leaving 3 subgoals.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Claim L2: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2
Let x4 of type ι → ο be given.
set y5 to be λ x5 . x4
Apply minus_SNo_invol with
y2,
λ x6 x7 . y5 (add_SNo x6 (minus_SNo y3)) (add_SNo x7 (minus_SNo y3)) leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Let x4 of type ι → ι → ο be given.
Apply L2 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H3: x4 y3 y3.
The subproof is completed by applying H3.