Let x0 of type ι → ι be given.
Apply nat_ind with
λ x1 . (∀ x2 . x2 ∈ x1 ⟶ x0 x2 ∈ omega) ⟶ 05ecb.. x0 x1 ∈ omega leaving 2 subgoals.
Assume H0:
∀ x1 . x1 ∈ 0 ⟶ x0 x1 ∈ omega.
Apply unknownprop_89e7310d38716ec0d15b566dbd8df2f84011da8cd7b706cd43ff87121048033c with
x0,
λ x1 x2 . x2 ∈ omega.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Let x1 of type ι be given.
Apply unknownprop_bfe386a724e0556e84046f452d416531498b4ec738b589b2f6a1e7f84e7dc85a with
x0,
x1,
λ x2 x3 . x3 ∈ omega leaving 2 subgoals.
The subproof is completed by applying H0.
Apply mul_SNo_In_omega with
05ecb.. x0 x1,
x0 x1 leaving 2 subgoals.
Apply H1.
Let x2 of type ι be given.
Assume H3: x2 ∈ x1.
Apply H2 with
x2.
Apply ordsuccI1 with
x1,
x2.
The subproof is completed by applying H3.
Apply H2 with
x1.
The subproof is completed by applying ordsuccI2 with x1.