Let x0 of type ο be given.
Apply H1 with
λ x1 . binunion {mul_SNo 2 x2|x2 ∈ omega,x2 ∈ x1} {add_SNo (mul_SNo 2 x2) 1|x2 ∈ omega,(λ x3 . SetAdjoin x3 (Sing 1)) x2 ∈ x1}.
Apply andI with
∀ x1 . x1 ∈ SNoS_ (ordsucc omega) ⟶ (λ x2 . binunion {mul_SNo 2 x3|x3 ∈ omega,x3 ∈ x2} {add_SNo (mul_SNo 2 x3) 1|x3 ∈ omega,(λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x2}) x1 ∈ prim4 omega,
∀ x1 . x1 ∈ SNoS_ (ordsucc omega) ⟶ ∀ x2 . x2 ∈ SNoS_ (ordsucc omega) ⟶ (λ x3 . binunion {mul_SNo 2 x4|x4 ∈ omega,x4 ∈ x3} {add_SNo (mul_SNo 2 x4) 1|x4 ∈ omega,(λ x5 . SetAdjoin x5 (Sing 1)) x4 ∈ x3}) x1 = (λ x3 . binunion {mul_SNo 2 x4|x4 ∈ omega,x4 ∈ x3} {add_SNo (mul_SNo 2 x4) 1|x4 ∈ omega,(λ x5 . SetAdjoin x5 (Sing 1)) x4 ∈ x3}) x2 ⟶ x1 = x2 leaving 2 subgoals.
Let x1 of type ι be given.
Apply PowerI with
omega,
(λ x2 . binunion {mul_SNo 2 x3|x3 ∈ omega,x3 ∈ x2} {add_SNo (mul_SNo 2 x3) 1|x3 ∈ omega,(λ x4 . SetAdjoin x4 (Sing 1)) x3 ∈ x2}) x1.
Apply binunion_Subq_min with
{mul_SNo 2 x2|x2 ∈ omega,x2 ∈ x1},
{add_SNo (mul_SNo 2 x2) 1|x2 ∈ omega,(λ x3 . SetAdjoin x3 (Sing 1)) x2 ∈ x1},
omega leaving 2 subgoals.
Let x2 of type ι be given.
Apply ReplSepE_impred with
omega,
λ x3 . x3 ∈ x1,
λ x3 . mul_SNo 2 x3,
x2,
x2 ∈ omega leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H5: x3 ∈ x1.
Apply H6 with
λ x4 x5 . x5 ∈ omega.
Apply mul_SNo_In_omega with
2,
x3 leaving 2 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
The subproof is completed by applying H4.
Let x2 of type ι be given.