English
For a fixed function g: G → β with symmetry condition, the value g at mulIndicator (s Δ t) f x equals g applied to the quotient of the indicators over s and t. This expresses how mulIndicator interacts with symmetric difference.
Русский
При симметрической разности s и t значение g на mulIndicator (s Δ t) f x равно g прикладному делению показателей на s и t.
LaTeX
$$$$ \\forall g:\\, G \\to \\beta,\\ \\forall s,t:\\ Set\\,\\alpha, f:\\alpha \\to G, x:\\alpha,\\quad g( mulIndicator(s \\Delta t) f x ) = g( mulIndicator s f x / mulIndicator t f x ). $$$$
Lean4
@[to_additive]
theorem apply_mulIndicator_symmDiff {g : G → β} (hg : ∀ x, g x⁻¹ = g x) (s t : Set α) (f : α → G) (x : α) :
g (mulIndicator (s ∆ t) f x) = g (mulIndicator s f x / mulIndicator t f x) := by
by_cases hs : x ∈ s <;> by_cases ht : x ∈ t <;> simp [mem_symmDiff, *]
