Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoLeE with
0,
x0,
mul_SNo x0 x0 = mul_SNo x1 x1 ⟶ x0 = x1 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply SNoLeE with
0,
x1,
mul_SNo x0 x0 = mul_SNo x1 x1 ⟶ x0 = x1 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply SNo_pos_sqr_uniq with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Assume H5: 0 = x1.
Apply H5 with
λ x2 x3 . mul_SNo x0 x0 = mul_SNo x2 x2 ⟶ x0 = x2.
Apply mul_SNo_zeroR with
0,
λ x2 x3 . mul_SNo x0 x0 = x3 ⟶ x0 = 0 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_zero_or_sqr_pos with
x0,
x0 = 0 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H7: x0 = 0.
The subproof is completed by applying H7.
Apply FalseE with
x0 = 0.
Apply SNoLt_irref with
0.
Apply H6 with
λ x2 x3 . SNoLt 0 x2.
The subproof is completed by applying H7.
Assume H4: 0 = x0.
Apply H4 with
λ x2 x3 . mul_SNo x2 x2 = mul_SNo x1 x1 ⟶ x2 = x1.
Apply mul_SNo_zeroR with
0,
λ x2 x3 . x3 = mul_SNo x1 x1 ⟶ 0 = x1 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_zero_or_sqr_pos with
x1,
0 = x1 leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H6: x1 = 0.
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying H6 with λ x3 x4 . x2 x4 x3.
Apply FalseE with
0 = x1.
Apply SNoLt_irref with
0.
Apply H5 with
λ x2 x3 . SNoLt 0 x3.
The subproof is completed by applying H6.