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.

Let x4 of type ι be given.

Let x5 of type ι be given.

Assume H0: x0 ∈ real.

Assume H1: x1 ∈ real.

Assume H2: ∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0.

Assume H3: ∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6)).

Assume H4: ∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6).

Assume H5: ∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0.

Assume H6: ∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6).

Assume H7: ∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7).

Assume H8: x5 ∈ setexp (SNoS_ omega) omega.

Assume H9: ∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1.

Assume H10: ∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6)).

Assume H11: ∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6).

Assume H12: ∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1.

Assume H13: ∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6).

Assume H14: ∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7).

Assume H15: SNo x0.

Assume H16: SNo x1.

Assume H17: SNo (add_SNo x0 x1).

Assume H18: ∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0.

Assume H19: ∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1.

Assume H20: ∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega.

Assume H21: ∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6)).

Assume H22: ∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega.

Assume H23: ∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6)).

Apply real_add_SNo with x0, x1 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

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Proofgold Proof