Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with
x2.
The subproof is completed by applying H2.
Apply add_SNo_assoc with
x0,
minus_SNo x1,
add_SNo x1 (minus_SNo x2),
λ x3 x4 . add_SNo x0 (minus_SNo x2) = x3 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.
Apply SNo_add_SNo with
x1,
minus_SNo x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L4.
Claim L5: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
set y6 to be λ x6 . x5
Apply add_SNo_minus_L2 with
y3,
minus_SNo y4,
λ x8 x9 . y7 x9 x8 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L4.
The subproof is completed by applying H5.
Let x5 of type ι → ι → ο be given.
Apply L5 with
λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H6: x5 y4 y4.
The subproof is completed by applying H6.