mono — Mathlib

English

If f is upper semicontinuous within s at x, and t ⊆ s, then f is upper semicontinuous within t at x.

Русский

Если f верхняя полупрерывность внутри s в x, и t ⊆ s, то f верхняя полупрерывность внутри t в x.

LaTeX

$$$\\displaystyle \\text{UpperSemicontinuousWithinAt}(f,t,x) \\text{ при } t \\subseteq s \\text{ от } \\text{UpperSemicontinuousWithinAt}(f,s,x)$$$

Lean4

theorem mono (h : UpperSemicontinuousWithinAt f s x) (hst : t ⊆ s) : UpperSemicontinuousWithinAt f t x := fun y hy =>
  Filter.Eventually.filter_mono (nhdsWithin_mono _ hst) (h y hy)

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