Prove2Me

`BookProof.EnergyBandDecomposition.energy_sub_scalar_lt` (hε : 0 < ε) (hx : x ∈ band E ε k) : |E x - k * ε| < ε

Proved
BookProof.EnergyBandDecomposition.energy_sub_scalar_lt

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterEnergyBandDecomposition.

BookProof.EnergyBandDecomposition.energy_sub_scalar_lt (hε : 0 < ε) (hx : x ∈ band E ε k) : |E x - k * ε| < ε

Formalization note: Lean 4 identifier BookProof.EnergyBandDecomposition.energy_sub_scalar_lt.

Preamble
-- Generated from ChapterEnergyBandDecomposition.lean — theorem BookProof.EnergyBandDecomposition.energy_sub_scalar_lt
import Mathlib
import Definitions.Def_ChapterEnergyBandDecomposition
open BookProof.EnergyBandDecomposition



open MeasureTheory

variable {X : Type*} {E : X → ℝ} {ε : ℝ} {k : ℤ} {x : X}
Formal statement
theorem BookProof.EnergyBandDecomposition.energy_sub_scalar_lt (hε : 0 < ε) (hx : x ∈ band E ε k) :
    |E x - k * ε| < ε := by sorry

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