Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoS_E2 with
omega,
x0,
add_SNo x0 x1 ∈ SNoS_ omega leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H0.
Apply SNoS_E2 with
omega,
x1,
add_SNo x0 x1 ∈ SNoS_ omega leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H1.
Apply SNoS_I with
omega,
add_SNo x0 x1,
SNoLev (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
Apply SNoLev_ordinal with
add_SNo x0 x1.
Apply SNo_add_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Apply ordinal_In_Or_Subq with
SNoLev (add_SNo x0 x1),
omega,
SNoLev (add_SNo x0 x1) ∈ omega leaving 4 subgoals.
The subproof is completed by applying L10.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H11.
Apply In_irref with
add_SNo (SNoLev x0) (SNoLev x1),
SNoLev (add_SNo x0 x1) ∈ omega.
Apply add_SNo_Lev_bd with
x0,
x1,
add_SNo (SNoLev x0) (SNoLev x1) leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Apply H11 with
add_SNo (SNoLev x0) (SNoLev x1).
Apply add_SNo_In_omega with
SNoLev x0,
SNoLev x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply SNoLev_ with
add_SNo x0 x1.
Apply SNo_add_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H8.