equicontinuousWithinAt — Mathlib

English

If H is equicontinuous at x0 within T (EquicontinuousWithinAt H T x0) and S ⊆ T, then H is equicontinuous at x0 within S.

Русский

Если H эквиконтинуально в точке x0 внутри T (EquicontinuousWithinAt H T x0) и S ⊆ T, то H эквиконтинуально в точке x0 внутри S.

LaTeX

$$EquicontinuousWithinAt(F,T,x0) ∧ S ⊆ T ⇒ EquicontinuousWithinAt(F,S,x0)$$

Lean4

theorem equicontinuousWithinAt {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀) (S : Set X) :
    EquicontinuousWithinAt F S x₀ := fun U hU ↦ (H U hU).filter_mono inf_le_left

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