English
Let Φ ⊆ M and suppose f x = 2 and MapsTo (reflection h) Φ Φ. Then the reflection is an involution on Φ, i.e., Φ is invariant under reflection h.
Русский
Пусть Φ ⊆ M и f x = 2 и MapsTo (reflection h) Φ Φ. Тогда отражение действует как инволюция на Φ, т.е. Φ инвариантно относительно reflection h.
LaTeX
$$$\\text{If }h:\\ f(x)=2\\text{ and }MapsTo(\\mathrm{reflection}\\ h,\\ Φ,Φ),\\text{ then }\\mathrm{End.invtSubmodule}(\\mathrm{reflection}\\ h)\\text{ holds on }Φ.$$$
Lean4
theorem invOn_reflection_of_mapsTo {Φ : Set M} (h : f x = 2) : InvOn (reflection h) (reflection h) Φ Φ :=
⟨fun x _ ↦ involutive_reflection h x, fun x _ ↦ involutive_reflection h x⟩