Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply add_SNo_assoc with
x0,
x1,
x2,
λ x3 x4 . x3 = add_SNo (add_SNo x0 x2) x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply add_SNo_assoc with
x0,
x2,
x1,
λ x3 x4 . add_SNo x0 (add_SNo x1 x2) = x3 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Claim L3: ∀ 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_com with
y3,
y4,
λ x7 x8 . y6 (add_SNo x2 x7) (add_SNo x2 x8) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
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.