Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Claim L3: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
Apply minus_add_SNo_distr with
x2,
add_SNo y3 y4,
λ x6 . x5 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with
y3,
y4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Claim L4: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
set y9 to be λ x9 . x8
Apply minus_add_SNo_distr with
x5,
y6,
λ x10 x11 . y9 (add_SNo (minus_SNo y4) x10) (add_SNo (minus_SNo y4) x11) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
set y8 to be λ x8 . y7
Apply L4 with
λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.
Assume H5: y8 y7 y7.
The subproof is completed by applying H5.
The subproof is completed by applying L4.
Let x5 of type ι → ι → ο be given.
Apply L3 with
λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H4: x5 y4 y4.
The subproof is completed by applying H4.