circleAverage_sub — Mathlib

English

If f is constant with value a, then circleAverage f c R = a.

Русский

Если f константа со значением a, то circleAverage f c R = a.

LaTeX

$$$circleAverage\ (\lambda\_\to a)\ c\ R = a$$$

Lean4

/-- Circle averages commute with subtraction. -/
theorem circleAverage_sub (hf₁ : CircleIntegrable f₁ c R) (hf₂ : CircleIntegrable f₂ c R) :
    circleAverage (f₁ - f₂) c R = circleAverage f₁ c R - circleAverage f₂ c R :=
  by
  rw [circleAverage, circleAverage, circleAverage, ← smul_sub]
  congr
  apply intervalIntegral.integral_sub hf₁ hf₂

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