Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply H0 with
SNoLe (SNoCut x0 x1) (SNoCut x2 x3).
Assume H4:
and (∀ x4 . x4 ∈ x0 ⟶ SNo x4) (∀ x4 . x4 ∈ x1 ⟶ SNo x4).
Apply H4 with
(∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 ⟶ SNoLt x4 x5) ⟶ SNoLe (SNoCut x0 x1) (SNoCut x2 x3).
Assume H5:
∀ x4 . x4 ∈ x0 ⟶ SNo x4.
Assume H6:
∀ x4 . x4 ∈ x1 ⟶ SNo x4.
Assume H7:
∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 ⟶ SNoLt x4 x5.
Apply H1 with
SNoLe (SNoCut x0 x1) (SNoCut x2 x3).
Assume H8:
and (∀ x4 . x4 ∈ x2 ⟶ SNo x4) (∀ x4 . x4 ∈ x3 ⟶ SNo x4).
Apply H8 with
(∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x3 ⟶ SNoLt x4 x5) ⟶ SNoLe (SNoCut x0 x1) (SNoCut x2 x3).
Assume H9:
∀ x4 . x4 ∈ x2 ⟶ SNo x4.
Assume H10:
∀ x4 . x4 ∈ x3 ⟶ SNo x4.
Assume H11:
∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x3 ⟶ SNoLt x4 x5.
Apply SNoCutP_SNoCut with
x0,
x1,
SNoLe (SNoCut x0 x1) (SNoCut x2 x3) leaving 2 subgoals.
The subproof is completed by applying H0.