Let x0 of type ι → ο be given.
Assume H0:
∀ x1 . x1 ∈ omega ⟶ x0 x1.
Let x1 of type ι be given.
Apply binunionE with
omega,
{minus_CSNo x2|x2 ∈ omega},
x1,
x0 x1 leaving 3 subgoals.
The subproof is completed by applying H2.
Apply H0 with
x1.
The subproof is completed by applying H3.
Apply ReplE_impred with
omega,
minus_CSNo,
x1,
x0 x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Apply minus_SNo_minus_CSNo with
x2,
λ x3 x4 . x1 = x3 ⟶ x0 x1 leaving 2 subgoals.
Apply ordinal_SNo with
x2.
Apply nat_p_ordinal with
x2.
Apply omega_nat_p with
x2.
The subproof is completed by applying H4.
Apply H5 with
λ x3 x4 . x0 x4.
Apply H1 with
x2.
The subproof is completed by applying H4.