one_div_mul_sub_mul_one_div_eq_one_div_add_one_div

English

If a and b are nonzero in a semifield, then 1/a · (b − a) · 1/b = 1/a − 1/b.

Русский

Если a и b не равны нулю в полем поле, то 1/a · (b − a) · 1/b = 1/a − 1/b.

LaTeX

$$$\\frac{1}{a} \\cdot (b - a) \\cdot \\frac{1}{b} = \\frac{1}{a} - \\frac{1}{b}, \\quad a \\neq 0, \\ b \\neq 0$$$

Lean4

theorem one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb : b ≠ 0) :
    1 / a * (b - a) * (1 / b) = 1 / a - 1 / b := by simpa only [one_div] using (inv_sub_inv' ha hb).symm

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