Proofgold Signed Transaction

Param real^real : ι
Param setexp^setexp : ι → ι → ι
Param SNoS_^SNoS_ : ι → ι
Param omega^omega : ι
Param SNoLt^SNoLt : ι → ι → ο
Param ap^ap : ι → ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param SNo^SNo : ι → ο
Param abs_SNo^abs_SNo : ι → ι
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Theorem 34bd0..^{Conj_real_add_SNo__44__7} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x2 ∈ setexp (SNoS_ omega) omega ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x4 ∈ setexp (SNoS_ omega) omega ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ add_SNo x0 x1 ∈ real (proof)
Param ordsucc^ordsucc : ι → ι
Theorem c4ebc..^{Conj_real_add_SNo__43__10} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x2 ∈ setexp (SNoS_ omega) omega ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 59aa9..^{Conj_real_add_SNo__40__19} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem bce5d..^{Conj_real_add_SNo__41__9} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem d6042..^{Conj_real_add_SNo__40__2} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 786e0..^{Conj_real_add_SNo__37__9} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 5c476..^{Conj_real_add_SNo__36__12} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x5 (ordsucc x6))) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem ec53b..^{Conj_real_add_SNo__36__15} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x5 (ordsucc x6))) ⟶ add_SNo x0 x1 ∈ real (proof)
Param SNoLev^SNoLev : ι → ι
Param setprod^setprod : ι → ι → ι
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Theorem 7cb2e..^{Conj_mul_SNo_eq__25__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . SNo x0 ⟶ SNo x1 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x5 . SNo x5 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)} ⟶ SNoCut (binunion {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)}) = SNoCut (binunion {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)}) (proof)
Theorem 7e49d..^{Conj_mul_SNo_eq__25__0} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . SNo x1 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x5 . SNo x5 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x4 = x3 x0 x4) ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} ⟶ {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)} = {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)} ⟶ SNoCut (binunion {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (x2 (ap x5 0) x1) (add_SNo (x2 x0 (ap x5 1)) (minus_SNo (x2 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)}) = SNoCut (binunion {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (x3 (ap x5 0) x1) (add_SNo (x3 x0 (ap x5 1)) (minus_SNo (x3 (ap x5 0) (ap x5 1))))|x5 ∈ setprod (SNoR x0) (SNoL x1)}) (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Theorem 98ac0..^{Conj_KnasterTarski_set__3__0} : ∀ x0 . ∀ x1 : ι → ι . {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ∈ prim4 x0 ⟶ x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ∈ prim4 x0 ⟶ (∀ x2 . x2 ∈ prim4 x0 ⟶ x1 x2 ⊆ x2 ⟶ {x3 ∈ x0|∀ x4 . x4 ∈ prim4 x0 ⟶ x1 x4 ⊆ x4 ⟶ x3 ∈ x4} ⊆ x2) ⟶ x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ⊆ {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ⟶ x1 (x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3}) ⊆ x1 {x2 ∈ x0|∀ x3 . x3 ∈ prim4 x0 ⟶ x1 x3 ⊆ x3 ⟶ x2 ∈ x3} ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ prim4 x0) (x1 x3 = x3) ⟶ x2) ⟶ x2 (proof)
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem 24d2c..^{Conj_PigeonHole_nat__1__0} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 x4 . x3 ∈ ordsucc (ordsucc x0) ⟶ ordsucc x4 ∈ ordsucc (ordsucc x0) ⟶ not (x2 ⊆ x3) ⟶ x2 ⊆ x4 ⟶ x1 x3 = x1 (ordsucc x4) ⟶ x3 = ordsucc x4 ⟶ ∀ x5 : ο . x5 (proof)
Param If_i^If_i : ο → ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Theorem 814b8..^{Conj_PigeonHole_nat_bij__2__2} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 x4 . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 = x1 x6 ⟶ x5 = x6) ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x1 x6 = x2) ⟶ x5) ⟶ x5) ⟶ x4 ∈ ordsucc x0 ⟶ ((x3 = x0 ⟶ ∀ x5 : ο . x5) ⟶ x3 ∈ x0) ⟶ If_i (x3 = x0) x2 (x1 x3) = If_i (x4 = x0) x2 (x1 x4) ⟶ x3 = x4 (proof)
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Definition ordinal^ordinal := λ x0 . and (TransSet x0) (∀ x1 . x1 ∈ x0 ⟶ TransSet x1)
Param PNo_strict_imv^{PNo_strict_imv} : (ι → (ι → ο) → ο) → (ι → (ι → ο) → ο) → ι → (ι → ο) → ο
Param PNo_strict_lowerbd^{PNo_strict_lowerbd} : (ι → (ι → ο) → ο) → ι → (ι → ο) → ο
Param PNo_strict_upperbd^{PNo_strict_upperbd} : (ι → (ι → ο) → ο) → ι → (ι → ο) → ο
Param iff^iff : ο → ο → ο
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem d71fb..^{Conj_PNo_strict_imv_pred_eq__6__3} : ∀ x0 x1 : ι → (ι → ο) → ο . ∀ x2 . ∀ x3 x4 : ι → ο . ordinal x2 ⟶ TransSet x2 ⟶ (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 : ι → ο . not (PNo_strict_imv x0 x1 x5 x6)) ⟶ PNo_strict_lowerbd x1 x2 x3 ⟶ PNo_strict_upperbd x0 x2 x4 ⟶ PNo_strict_lowerbd x1 x2 x4 ⟶ (∀ x5 . ordinal x5 ⟶ x5 ∈ x2 ⟶ iff (x3 x5) (x4 x5)) ⟶ ∀ x5 . x5 ∈ x2 ⟶ iff (x3 x5) (x4 x5) (proof)
Param PNo_bd^PNo_bd : (ι → (ι → ο) → ο) → (ι → (ι → ο) → ο) → ι
Theorem 33d61..^{Conj_PNo_bd_In__1__3} : ∀ x0 x1 : ι → (ι → ο) → ο . ∀ x2 x3 . ∀ x4 : ι → ο . (∀ x5 . x5 ∈ PNo_bd x0 x1 ⟶ ∀ x6 : ι → ο . not (PNo_strict_imv x0 x1 x5 x6)) ⟶ x3 ∈ ordsucc x2 ⟶ PNo_strict_imv x0 x1 x3 x4 ⟶ not (x3 ∈ PNo_bd x0 x1) (proof)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem 79465..^{Conj_SNo_etaE__5__1} : ∀ x0 x1 x2 . SNoLt x0 x1 ⟶ SNo x1 ⟶ SNoLev x1 = x2 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ and (and (SNo x1) (SNoLev x1 ∈ SNoLev x0)) (SNoLt x0 x1) (proof)
Param SNo_rec_i^SNo_rec_i : (ι → (ι → ι) → ι) → ι → ι
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Theorem 85918..^{Conj_SNo_rec2_eq__1__1} : ∀ x0 : ι → ι → (ι → ι → ι) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ x5 . SNo x5 ⟶ ∀ x6 . SNo x6 ⟶ ∀ x7 x8 : ι → ι → ι . (∀ x9 . x9 ∈ SNoS_ (SNoLev x5) ⟶ ∀ x10 . SNo x10 ⟶ x7 x9 x10 = x8 x9 x10) ⟶ (∀ x9 . x9 ∈ SNoS_ (SNoLev x6) ⟶ x7 x5 x9 = x8 x5 x9) ⟶ x0 x5 x6 x7 = x0 x5 x6 x8) ⟶ (∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ x2 x5 = x3 x5) ⟶ SNo x4 ⟶ (∀ x5 . ordinal x5 ⟶ ∀ x6 . x6 ∈ SNoS_ x5 ⟶ SNo_rec_i (λ x8 . λ x9 : ι → ι . x0 x1 x8 (λ x10 x11 . If_i (x10 = x1) (x9 x11) (x2 x10 x11))) x6 = SNo_rec_i (λ x8 . λ x9 : ι → ι . x0 x1 x8 (λ x10 x11 . If_i (x10 = x1) (x9 x11) (x3 x10 x11))) x6) ⟶ SNo_rec_i (λ x6 . λ x7 : ι → ι . x0 x1 x6 (λ x8 x9 . If_i (x8 = x1) (x7 x9) (x2 x8 x9))) x4 = SNo_rec_i (λ x6 . λ x7 : ι → ι . x0 x1 x6 (λ x8 x9 . If_i (x8 = x1) (x7 x9) (x3 x8 x9))) x4 (proof)
Param SNoCutP^SNoCutP : ι → ι → ο
Theorem e0a4a..^{Conj_minus_SNo_prop1__2__2} : ∀ x0 x1 . SNo x0 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ and (and (and (SNo (minus_SNo x2)) (∀ x3 . x3 ∈ SNoL x2 ⟶ SNoLt (minus_SNo x2) (minus_SNo x3))) (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (minus_SNo x3) (minus_SNo x2))) (SNoCutP (prim5 (SNoR x2) minus_SNo) (prim5 (SNoL x2) minus_SNo))) ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ x1 ∈ SNoS_ (SNoLev x0) ⟶ and (and (SNo (minus_SNo x1)) (∀ x2 . x2 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2))) (∀ x2 . x2 ∈ SNoR x1 ⟶ SNoLt (minus_SNo x2) (minus_SNo x1)) (proof)
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Theorem cf6dd..^{Conj_minus_SNo_prop1__9__3} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ and (and (and (SNo (minus_SNo x1)) (∀ x2 . x2 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2))) (∀ x2 . x2 ∈ SNoR x1 ⟶ SNoLt (minus_SNo x2) (minus_SNo x1))) (SNoCutP (prim5 (SNoR x1) minus_SNo) (prim5 (SNoL x1) minus_SNo))) ⟶ (∀ x1 . x1 ∈ SNoL x0 ⟶ and (and (SNo (minus_SNo x1)) (∀ x2 . x2 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2))) (∀ x2 . x2 ∈ SNoR x1 ⟶ SNoLt (minus_SNo x2) (minus_SNo x1))) ⟶ SNoCutP (prim5 (SNoR x0) minus_SNo) (prim5 (SNoL x0) minus_SNo) ⟶ and (and (and (SNo (minus_SNo x0)) (∀ x1 . x1 ∈ SNoL x0 ⟶ SNoLt (minus_SNo x0) (minus_SNo x1))) (∀ x1 . x1 ∈ SNoR x0 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0))) (SNoCutP (prim5 (SNoR x0) minus_SNo) (prim5 (SNoL x0) minus_SNo)) (proof)
Known minus_SNo_Lev_lem2^{minus_SNo_Lev_lem2} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) ⊆ SNoLev x0
Theorem 49f1d..^{Conj_minus_SNo_Lev_lem1__22__2} : ∀ x0 x1 . TransSet x0 ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ SNoS_ x2 ⟶ SNoLev (minus_SNo x3) ⊆ SNoLev x3) ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNoCutP (prim5 (SNoR x1) minus_SNo) (prim5 (SNoL x1) minus_SNo) ⟶ SNoLev (minus_SNo x1) ⊆ SNoLev x1 (proof)
Known add_SNo_ordinal_SL^{add_SNo_ordinal_SL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Theorem 4d514..^{Conj_add_SNo_ordinal_SL__14__0} : ∀ x0 x1 . ordinal x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ add_SNo (ordsucc x0) x2 = ordsucc (add_SNo x0 x2)) ⟶ SNo x0 ⟶ SNo x1 ⟶ ordinal (add_SNo x0 x1) ⟶ ordinal (ordsucc x0) ⟶ SNo (ordsucc x0) ⟶ add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1) (proof)
Theorem 36a11..^{Conj_mul_SNo_eq__19__0} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem e4f91..^{Conj_mul_SNo_eq__20__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 36a11..^{Conj_mul_SNo_eq__19__0} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem e4f91..^{Conj_mul_SNo_eq__20__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 36a11..^{Conj_mul_SNo_eq__19__0} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem e4f91..^{Conj_mul_SNo_eq__20__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 3816f..^{Conj_mul_SNo_eq__22__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . SNo x1 ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ x5 ∈ SNoS_ (SNoLev x1) ⟶ SNo x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Known SNoR_SNoS_^SNoR_SNoS_ : ∀ x0 . SNoR x0 ⊆ SNoS_ (SNoLev x0)
Theorem a3487..^{Conj_mul_SNo_eq__18__4} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . SNo x0 ⟶ SNo x1 ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x5 ∈ SNoR x1 ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 36a11..^{Conj_mul_SNo_eq__19__0} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem e4f91..^{Conj_mul_SNo_eq__20__3} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x0 x6 = x3 x0 x6) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem 3cff7..^{Conj_mul_SNo_SNoR_interpolate__4__3} : ∀ x0 x1 x2 x3 x4 x5 . SNo x2 ⟶ SNo x3 ⟶ SNoLt x3 x2 ⟶ SNoLt (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x3 (mul_SNo x4 x5)) ⟶ SNoLt x2 x3 ⟶ SNoLt x3 (mul_SNo x0 x1) (proof)
Theorem 3cff7..^{Conj_mul_SNo_SNoR_interpolate__4__3} : ∀ x0 x1 x2 x3 x4 x5 . SNo x2 ⟶ SNo x3 ⟶ SNoLt x3 x2 ⟶ SNoLt (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x3 (mul_SNo x4 x5)) ⟶ SNoLt x2 x3 ⟶ SNoLt x3 (mul_SNo x0 x1) (proof)
Known double_eps_1^double_eps_1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x0 = add_SNo x1 x2 ⟶ x0 = mul_SNo (eps_ 1) (add_SNo x1 x2)
Theorem 7bd1d..^{Conj_double_eps_1__1__1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x2 ⟶ add_SNo x0 x0 = add_SNo x1 x2 ⟶ SNo (add_SNo x1 x2) ⟶ x0 = mul_SNo (eps_ 1) (add_SNo x1 x2) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Theorem fb18d..^{Conj_SNo_approx_real_rep__1__0} : ∀ x0 x1 . (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoS_ omega ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0)) ⟶ not (∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLe (add_SNo x0 (eps_ x3)) x1) ⟶ x2) ⟶ x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Theorem 3aca5..^{Conj_real_add_SNo__23__14} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x4 (ordsucc x7)))) x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap (lam omega (λ x7 . add_SNo (ap x3 (ordsucc x7)) (ap x5 (ordsucc x7)))) x6)) ⟶ lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6))) ∈ setexp (SNoS_ omega) omega ⟶ lam omega (λ x6 . add_SNo (ap x3 (ordsucc x6)) (ap x5 (ordsucc x6))) ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x4 (ordsucc x7)))) x6) (add_SNo x0 x1)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo x0 x1) (add_SNo (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x4 (ordsucc x7)))) x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap (lam omega (λ x8 . add_SNo (ap x2 (ordsucc x8)) (ap x4 (ordsucc x8)))) x7) (ap (lam omega (λ x8 . add_SNo (ap x2 (ordsucc x8)) (ap x4 (ordsucc x8)))) x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo x0 x1) (ap (lam omega (λ x7 . add_SNo (ap x3 (ordsucc x7)) (ap x5 (ordsucc x7)))) x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap (lam omega (λ x8 . add_SNo (ap x3 (ordsucc x8)) (ap x5 (ordsucc x8)))) x6) (ap (lam omega (λ x8 . add_SNo (ap x3 (ordsucc x8)) (ap x5 (ordsucc x8)))) x7)) ⟶ SNoCutP (prim5 omega (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6)))))) (prim5 omega (ap (lam omega (λ x6 . add_SNo (ap x3 (ordsucc x6)) (ap x5 (ordsucc x6)))))) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 56dcc..^{Conj_real_add_SNo__30__7} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x5 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap (lam omega (λ x8 . add_SNo (ap x2 (ordsucc x8)) (ap x4 (ordsucc x8)))) x6 = add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap (lam omega (λ x8 . add_SNo (ap x3 (ordsucc x8)) (ap x5 (ordsucc x8)))) x6 = add_SNo (ap x3 (ordsucc x6)) (ap x5 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x4 (ordsucc x7)))) x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap (lam omega (λ x7 . add_SNo (ap x3 (ordsucc x7)) (ap x5 (ordsucc x7)))) x6)) ⟶ lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6))) ∈ setexp (SNoS_ omega) omega ⟶ lam omega (λ x6 . add_SNo (ap x3 (ordsucc x6)) (ap x5 (ordsucc x6))) ∈ setexp (SNoS_ omega) omega ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 0e33e..^{Conj_real_add_SNo__33__2} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x5 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap (lam omega (λ x8 . add_SNo (ap x2 (ordsucc x8)) (ap x4 (ordsucc x8)))) x6 = add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap (lam omega (λ x8 . add_SNo (ap x3 (ordsucc x8)) (ap x5 (ordsucc x8)))) x6 = add_SNo (ap x3 (ordsucc x6)) (ap x5 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x4 (ordsucc x7)))) x6)) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem c87d9..^{Conj_real_add_SNo__35__13} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x2 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x2 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x4 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x4 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x3 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x3 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap x5 (ordsucc x6) ∈ SNoS_ omega) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNo (ap x5 (ordsucc x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ap (lam omega (λ x8 . add_SNo (ap x2 (ordsucc x8)) (ap x4 (ordsucc x8)))) x6 = add_SNo (ap x2 (ordsucc x6)) (ap x4 (ordsucc x6))) ⟶ add_SNo x0 x1 ∈ real (proof)