Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNo_div_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_div_SNo with
x0,
mul_SNo x1 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_mul_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply xm with
x1 = 0,
div_SNo (div_SNo x0 x1) x2 = div_SNo x0 (mul_SNo x1 x2) leaving 2 subgoals.
Assume H5: x1 = 0.
Apply H5 with
λ x3 x4 . div_SNo (div_SNo x0 x4) x2 = div_SNo x0 (mul_SNo x4 x2).
Apply mul_SNo_zeroL with
x2,
λ x3 x4 . div_SNo (div_SNo x0 0) x2 = div_SNo x0 x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply div_SNo_0_denum with
x0,
λ x3 x4 . div_SNo x4 x2 = x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply div_SNo_0_num with
x2.
The subproof is completed by applying H2.
Assume H5: x1 = 0 ⟶ ∀ x3 : ο . x3.
Apply xm with
x2 = 0,
div_SNo (div_SNo x0 x1) x2 = div_SNo x0 (mul_SNo x1 x2) leaving 2 subgoals.
Assume H6: x2 = 0.
Apply H6 with
λ x3 x4 . div_SNo (div_SNo x0 x1) x4 = div_SNo x0 (mul_SNo x1 x4).
Apply mul_SNo_zeroR with
x1,
λ x3 x4 . div_SNo (div_SNo x0 x1) 0 = div_SNo x0 x4 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply div_SNo_0_denum with
x0,
λ x3 x4 . div_SNo (div_SNo x0 x1) 0 = x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply div_SNo_0_denum with
div_SNo x0 x1.
The subproof is completed by applying L3.
Assume H6: x2 = 0 ⟶ ∀ x3 : ο . x3.
Apply mul_SNo_nonzero_cancel with
x2,
div_SNo (div_SNo x0 x1) x2,
div_SNo x0 (mul_SNo x1 x2) leaving 5 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply SNo_div_SNo with
div_SNo x0 x1,
x2 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H2.
The subproof is completed by applying L4.
Apply mul_div_SNo_invR with
div_SNo x0 x1,
x2,
λ x3 x4 . x4 = mul_SNo x2 (div_SNo x0 (mul_SNo x1 x2)) leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply mul_SNo_nonzero_cancel with
x1,
div_SNo x0 x1,
mul_SNo x2 (div_SNo x0 (mul_SNo x1 x2)) leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying L3.
Apply SNo_mul_SNo with
x2,
div_SNo x0 (mul_SNo x1 x2) leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying L4.
Apply mul_div_SNo_invR with
x0,
x1,
λ x3 x4 . x4 = mul_SNo x1 (mul_SNo x2 (div_SNo x0 (mul_SNo x1 x2))) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
Apply mul_SNo_assoc with
x1,
x2,
div_SNo x0 (mul_SNo x1 x2),
λ x3 x4 . x0 = x4 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L4.
Let x3 of type ι → ι → ο be given.
Apply mul_div_SNo_invR with
x0,
mul_SNo x1 x2,
λ x4 x5 . x3 x5 x4 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_mul_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply H6.
Apply mul_SNo_nonzero_cancel with
x1,
x2,
0 leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H2.
The subproof is completed by applying SNo_0.
Apply mul_SNo_zeroR with
x1,
λ x4 x5 . mul_SNo x1 x2 = x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H7.