A work-rate equation becomes a quadratic identity

Proved

WorkbookSource.plus_525

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

lean-workbooksource-checked

Let $c$ be the number of days in wich worker $C$ does his duty. So $B$ does it in $b + c$ and $A$ in $a + b + c$ days. In one day each worker, $C$ , $B$ and $A$ , does respectively $\frac {1}{c}$ , $\frac {1}{b + c}$ and $\frac {1}{a + b + c}$ . So if $A$ and $B$ together do the duty in the same time $C$ does, we must have that: $\frac {1}{a + b + c} + \frac {1}{b + c} = \frac {1}{c}$ Which can be reduced to $c^2 = ab + b^2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_525 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_525  (a b c : ℝ)
  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
  (h₁ : 1 / (a + b + c) + 1 / (b + c) = 1 / c) :
  c^2 = a * b + b^2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_525; Apache-2.0

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