frequently_eq_iff_eventually_eq — Mathlib

English

For analytic f and g, either they are equal in a neighborhood of z0 or they are unequal on a punctured neighborhood around z0.

Русский

Для аналитических f и g либо они равны в окрестности z0, либо различаются на punctured окрестности z0.

LaTeX

$$$AnalyticAt 𝕜 f z0 \land AnalyticAt 𝕜 g z0 \Rightarrow (\exists cond \text{ or } \exists neq)$$$

Lean4

theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
    (∃ᶠ z in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ z in 𝓝 z₀, f z = g z := by
  simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)

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