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Proofgold Proof

pf
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.
Assume H0: SNoCutP x0 x1.
Assume H1: SNoCutP x2 x3.
Claim L2: ...
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Claim L3: ...
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Claim L4: ...
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Apply H0 with SNoCutP (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x0} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x2}) (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x1} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x3}).
Assume H5: and (∀ x4 . x4 ∈ x0 ⟶ SNo x4) (∀ x4 . x4 ∈ x1 ⟶ SNo x4).
Apply H5 with (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 ⟶ SNoLt x4 x5) ⟶ SNoCutP (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x0} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x2}) (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x1} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x3}).
Assume H6: ∀ x4 . x4 ∈ x0 ⟶ SNo x4.
Assume H7: ∀ x4 . x4 ∈ x1 ⟶ SNo x4.
Assume H8: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 ⟶ SNoLt x4 x5.
Apply H1 with SNoCutP (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x0} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x2}) (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x1} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x3}).
Assume H9: and (∀ x4 . x4 ∈ x2 ⟶ SNo x4) (∀ x4 . x4 ∈ x3 ⟶ SNo x4).
Apply H9 with (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x3 ⟶ SNoLt x4 x5) ⟶ SNoCutP (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x0} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x2}) (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x1} {add_SNo (SNoCut x0 x1) x4|x4 ∈ x3}).
Assume H10: ∀ x4 . x4 ∈ x2 ⟶ SNo x4.
Assume H11: ∀ x4 . x4 ∈ x3 ⟶ SNo x4.
Assume H12: ∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x3 ⟶ SNoLt x4 x5.
Apply and3I with ∀ x4 . x4 ∈ binunion {add_SNo x5 (SNoCut x2 x3)|x5 ∈ x0} {add_SNo (SNoCut x0 x1) x5|x5 ∈ x2} ⟶ SNo x4, ∀ x4 . x4 ∈ binunion {add_SNo x5 (SNoCut x2 x3)|x5 ∈ x1} {add_SNo (SNoCut x0 x1) ...|x5 ∈ x3} ⟶ SNo x4, ... leaving 3 subgoals.
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