Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply xm with
x1 = 0,
mul_SNo (div_SNo x0 x1) x2 = div_SNo (mul_SNo x0 x2) x1 leaving 2 subgoals.
Assume H3: x1 = 0.
Apply H3 with
λ x3 x4 . mul_SNo (div_SNo x0 x4) x2 = div_SNo (mul_SNo x0 x2) x4.
Apply div_SNo_0_denum with
x0,
λ x3 x4 . mul_SNo x4 x2 = div_SNo (mul_SNo x0 x2) 0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply div_SNo_0_denum with
mul_SNo x0 x2,
λ x3 x4 . mul_SNo 0 x2 = x4 leaving 2 subgoals.
Apply SNo_mul_SNo with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply mul_SNo_zeroL with
x2.
The subproof is completed by applying H2.
Assume H3: x1 = 0 ⟶ ∀ x3 : ο . x3.
Apply mul_SNo_nonzero_cancel with
x1,
mul_SNo (div_SNo x0 x1) x2,
div_SNo (mul_SNo x0 x2) x1 leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply SNo_mul_SNo with
div_SNo x0 x1,
x2 leaving 2 subgoals.
Apply SNo_div_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_div_SNo with
mul_SNo x0 x2,
x1 leaving 2 subgoals.
Apply SNo_mul_SNo with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply mul_div_SNo_invR with
mul_SNo x0 x2,
x1,
λ x3 x4 . mul_SNo x1 (mul_SNo (div_SNo x0 x1) x2) = x4 leaving 4 subgoals.
Apply SNo_mul_SNo with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply mul_SNo_assoc with
x1,
div_SNo x0 x1,
x2,
λ x3 x4 . x4 = mul_SNo x0 x2 leaving 4 subgoals.
The subproof is completed by applying H1.
Apply SNo_div_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι → ο be given.
Apply mul_div_SNo_invR with
x1,
x2,
λ x5 x6 . (λ x7 . x4) (mul_SNo x5 y3) (mul_SNo x6 y3) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Let x4 of type ι → ι → ο be given.
Apply L4 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H5: x4 y3 y3.
The subproof is completed by applying H5.