Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply mul_SNo_assoc with
x0,
x1,
x2,
λ x3 x4 . x3 = mul_SNo (mul_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 mul_SNo_assoc with
x0,
x2,
x1,
λ x3 x4 . mul_SNo x0 (mul_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.
Let x4 of type ι → ο be given.
Apply mul_SNo_com with
x2,
y3,
λ x5 x6 . (λ x7 . x4) (mul_SNo x1 x5) (mul_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.