Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 = 0 ⟶ ∀ x2 : ο . x2.
Assume H3: x1 = 0 ⟶ ∀ x2 : ο . x2.
Apply SNoLt_trichotomy_or_impred with
x0,
0,
False leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply SNoLt_trichotomy_or_impred with
x1,
0,
False leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
Apply SNoLt_irref with
0.
Apply H4 with
λ x2 x3 . SNoLt 0 x2.
Apply mul_SNo_neg_neg with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Assume H6: x1 = 0.
Apply H3.
The subproof is completed by applying H6.
Apply SNoLt_irref with
0.
Apply H4 with
λ x2 x3 . SNoLt x2 0.
Apply mul_SNo_neg_pos with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Assume H5: x0 = 0.
Apply H2.
The subproof is completed by applying H5.
Apply SNoLt_trichotomy_or_impred with
x1,
0,
False leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
Apply SNoLt_irref with
0.
Apply H4 with
λ x2 x3 . SNoLt x2 0.
Apply mul_SNo_pos_neg with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Assume H6: x1 = 0.
Apply H3.
The subproof is completed by applying H6.
Apply SNoLt_irref with
0.
Apply H4 with
λ x2 x3 . SNoLt 0 x2.
Apply mul_SNo_pos_pos with
x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H6.