Combining two fractions under a square root

Proved

WorkbookSource.base_1164

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that $\sqrt{2(\frac{a}{5-a}+\frac{b}{5-b})}=\sqrt{\frac{2[5(a+b)-2ab]}{25-5(a+b)+ab}}$

Source: InternLM Lean-Workbook, record lean_workbook_1164 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1164 (a b : ℝ): (a ≠ 5 ∧ b ≠ 5) → √(2 * (a / (5 - a) + b / (5 - b))) = √((2 * (5 * (a + b) - 2 * a * b)) / (25 - 5 * (a + b) + a * b))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1164; Apache-2.0

View graph