- t(i₁) i₂ > - t(i₂) i₁

Adding t(i₂) i₂ to both sides:

t(i₂) i₂ - t(i₁) i₂ > t(i₂) i₂ - t(i₂) i₁

[ t(i₂) - t(i₁) ] i₂ > t(i₂) [ i₂ - i₁ ]

Since i₂ > i₁ > 0, i₂ [ i₂ - i₁ ] is a positive quantity. Dividing both sides of the above inequality by that positive quantity:

In other words, MTR > ATR₂. Since ATR₂ > ATR₁, we conclude that the marginal tax rate is greater than the average tax rate at either endpoint.