Coefficients of a scaled law: `(x · L)_σ = x · L_σ`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Theorems.Thm_TypeDensity_lawInterp_lawScale
import Theorems.Thm_TypeDensity_lawInterp_coeff
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheory.ArithmeticFunction.Moebius
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.law_coeff_scale (x : Expr) (L : Law) (σ : FactorType) (q : ℚ) :
    (lawCoeff (lawScale x L) σ).eval q = x.eval q * (lawCoeff L σ).eval q := by sorry