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Lean-Workbook Plus 64095: Trigonometric identity

Proved
WorkbookRestored.plus_64095

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

If x=rcos⁡acos⁡bcos⁡cx=r\cos a\cos b\cos c, y=rcos⁡acos⁡bsin⁡cy=r\cos a\cos b\sin c, z=rsin⁡acos⁡bz=r\sin a\cos b, and u=rsin⁡bu=r\sin b, then x2+y2+z2+u2=r2x^2+y^2+z^2+u^2=r^2.

Source: Lean-Workbook row lean_workbook_plus_64095 (Apache-2.0). The unchanged proposition comes from the original record; its missing mathematical imports and namespaces are restored here.

Preamble
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
open Real
Formal statement
theorem WorkbookRestored.plus_64095 (x y z u r a b c : ℝ) : 
  x = r * cos a * cos b * cos c ∧ 
  y = r * cos a * cos b * sin c ∧ 
  z = r * sin a * cos b ∧ 
  u = r * sin b → 
  x^2 + y^2 + z^2 + u^2 = r^2   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_64095; immutable original Prove2Me node b33fc46d-e32d-4614-b17c-e5033204f428

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