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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0 ∈ real.
Assume H1: SNoLt 0 x0.
Apply real_E with x0, ∃ x1 . and (x1 ∈ real) (mul_SNo x0 x1 = 1) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2: SNo x0.
Assume H3: SNoLev x0 ∈ ordsucc omega.
Assume H4: x0 ∈ SNoS_ (ordsucc omega).
Assume H5: SNoLt (minus_SNo omega) x0.
Assume H6: SNoLt x0 omega.
Assume H7: ∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0.
Assume H8: ∀ x1 . x1 ∈ omega ⟶ ∃ x2 . and (x2 ∈ SNoS_ omega) (and (SNoLt x2 x0) (SNoLt x0 (add_SNo x2 (eps_ x1)))).
Apply SNoS_ordsucc_omega_bdd_eps_pos with x0, ∃ x1 . and (x1 ∈ real) (mul_SNo x0 x1 = 1) leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Let x1 of type ι be given.
Assume H9: (λ x2 . and (x2 ∈ omega) (SNoLt (mul_SNo (eps_ x2) x0) 1)) x1.
Apply H9 with ∃ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1).
Assume H10: x1 ∈ omega.
Assume H11: SNoLt (mul_SNo (eps_ x1) x0) 1.
Apply pos_small_real_recip_ex with mul_SNo (eps_ x1) x0, ∃ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) leaving 4 subgoals.
Apply real_mul_SNo with eps_ x1, x0 leaving 2 subgoals.
Apply SNoS_omega_real with eps_ x1.
Apply SNo_eps_SNoS_omega with x1.
The subproof is completed by applying H10.
The subproof is completed by applying H0.
Apply mul_SNo_pos_pos with eps_ x1, x0 leaving 4 subgoals.
Apply SNo_eps_ with x1.
The subproof is completed by applying H10.
The subproof is completed by applying H2.
Apply SNo_eps_pos with x1.
The subproof is completed by applying H10.
The subproof is completed by applying H1.
The subproof is completed by applying H11.
Let x2 of type ι be given.
Assume H12: (λ x3 . and (x3 ∈ real) (mul_SNo (mul_SNo (eps_ x1) x0) x3 = 1)) x2.
Apply H12 with ∃ x3 . and (x3 ∈ real) (mul_SNo x0 x3 = 1).
Assume H13: x2 ∈ real.
Assume H14: mul_SNo (mul_SNo (eps_ x1) x0) x2 = 1.
Let x3 of type ο be given.
Assume H15: ∀ x4 . and (x4 ∈ real) (mul_SNo x0 x4 = 1) ⟶ x3.
Apply H15 with mul_SNo (eps_ x1) x2.
Apply andI with mul_SNo (eps_ x1) x2 ∈ real, mul_SNo x0 (mul_SNo (eps_ x1) x2) = 1 leaving 2 subgoals.
Apply real_mul_SNo with eps_ x1, x2 leaving 2 subgoals.
Apply SNoS_omega_real with eps_ x1.
Apply SNo_eps_SNoS_omega with x1.
The subproof is completed by applying H10.
The subproof is completed by applying H13.
Apply mul_SNo_assoc with x0, eps_ x1, x2, λ x4 x5 . x5 = 1 leaving 4 subgoals.
The subproof is completed by applying H2.
Apply SNo_eps_ with x1.
The subproof is completed by applying H10.
Apply real_SNo with x2.
The subproof is completed by applying H13.
Apply mul_SNo_com with x0, eps_ x1, λ x4 x5 . mul_SNo x5 x2 = 1 leaving 3 subgoals.
The subproof is completed by applying H2.
Apply SNo_eps_ with x1.
The subproof is completed by applying H10.
The subproof is completed by applying H14.
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