Let x0 of type ι be given.
Let x1 of type ι be given.
Apply H1 with
∀ x2 . SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∃ x3 . and (x3 ∈ SNoR x2) (add_SNo x1 x3 = add_SNo x0 x0).
Apply H2 with
(∀ x2 . x2 ∈ SNoL x0 ⟶ SNo x2 ⟶ SNoLe x2 x1) ⟶ ∀ x2 . SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∃ x3 . and (x3 ∈ SNoR x2) (add_SNo x1 x3 = add_SNo x0 x0).
Apply SNoL_E with
x0,
x1,
∀ x2 . SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∃ x3 . and (x3 ∈ SNoR x2) (add_SNo x1 x3 = add_SNo x0 x0) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply SNoLev_ind with
λ x2 . SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∃ x3 . and (x3 ∈ SNoR x2) (add_SNo x1 x3 = add_SNo x0 x0).
Let x2 of type ι be given.
Apply SNoLt_SNoL_or_SNoR_impred with
add_SNo x1 x2,
add_SNo x0 x0,
∃ x3 . and (x3 ∈ SNoR x2) (add_SNo x1 x3 = add_SNo x0 x0) leaving 6 subgoals.
Apply SNo_add_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
The subproof is completed by applying L13.
The subproof is completed by applying H12.
Let x3 of type ι be given.
Apply SNoL_E with
add_SNo x0 x0,
x3,
∃ x4 . and (x4 ∈ SNoR x2) (add_SNo x1 x4 = add_SNo x0 x0) leaving 3 subgoals.
The subproof is completed by applying L13.
The subproof is completed by applying H17.