congr_of_insert — Mathlib

English

If h is ContDiffWithinAt on s at x, and s ∈ nhdsWithin x t, then h is ContDiffWithinAt on t at x.

Русский

Если h — ContDiffWithinAt на s в x, и s ∈ nhdsWithin x t, то h остаётся ContDiffWithinAt на t в x.

LaTeX

$$$h: ContDiffWithinAt 𝕜 n f s x \\rightarrow (s \\in nhdsWithin x t) \\rightarrow ContDiffWithinAt 𝕜 n f t x$$$

Lean4

theorem congr_of_insert (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ insert x s, f₁ y = f y) :
    ContDiffWithinAt 𝕜 n f₁ s x :=
  h.congr (fun y hy ↦ h₁ y (mem_insert_of_mem _ hy)) (h₁ x (mem_insert _ _))

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