Apply explicit_Nats_E with
{x0 ∈ real|natOfOrderedField_p real 0 1 add_SNo mul_SNo SNoLe x0},
0,
λ x0 . add_SNo x0 1,
explicit_Reals real 0 1 add_SNo mul_SNo SNoLe leaving 2 subgoals.
Apply explicit_Reals_I with
real,
0,
1,
add_SNo,
mul_SNo,
SNoLe leaving 3 subgoals.
The subproof is completed by applying explicit_OrderedField_real.
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H10:
and (SNoLe 0 x0) (0 = x0 ⟶ ∀ x2 : ο . x2).
Apply SNoLt_trichotomy_or_impred with
0,
x0,
SNoLt 0 x0 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
Apply real_SNo with
x0.
The subproof is completed by applying H8.
The subproof is completed by applying H11.
Assume H11: 0 = x0.
Apply FalseE with
SNoLt 0 x0.
Apply H10 with
False.
Assume H13: 0 = x0 ⟶ ∀ x2 : ο . x2.
Apply H13.
The subproof is completed by applying H11.
Apply FalseE with
SNoLt 0 x0.
Apply H10 with
False.
Assume H13: 0 = x0 ⟶ ∀ x2 : ο . x2.
Apply SNoLt_irref with
x0.
Apply SNoLtLe_tra with
x0,
0,
x0 leaving 5 subgoals.
Apply real_SNo with
x0.
The subproof is completed by applying H8.
The subproof is completed by applying SNo_0.
Apply real_SNo with
x0.
The subproof is completed by applying H8.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Apply L7 with
λ x2 x3 . SNoLe 0 x1 ⟶ ∃ x4 . and (x4 ∈ x3) (SNoLe x1 (mul_SNo x4 x0)).
Apply real_Archimedean with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying L11.