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,
minus_SNo x1,
add_SNo x1 x2,
λ x3 x4 . x3 = add_SNo x0 x2 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply SNo_add_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι → ο be given.
Apply add_SNo_minus_L2 with
x2,
y3,
λ x5 x6 . (λ x7 . x4) (add_SNo x1 x5) (add_SNo x1 x6) leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι → ι → ο be given.
Apply L3 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H4: x4 y3 y3.
The subproof is completed by applying H4.