Proof

Let us call [Maple Math] and [Maple Math] then [Maple Math] and [Maple Math] are polynomials of degree [Maple Math] or less. Hence, [Maple Math] must be a polynomial of degree [Maple Math] or less.

For [Maple Math] , we have that [Maple Math] and [Maple Math] . Therefore, [Maple Math] or [Maple Math] .

For [Maple Math] , [Maple Math] so, [Maple Math] or [Maple Math] .

Similarly, for [Maple Math] , we find that [Maple Math] .

Hence, [Maple Math] is a polynomial of degree [Maple Math] or less which fits the [Maple Math] data points ( [Maple Math] ) . Since such a polynomial is unique, [Maple Math] must be the Lagrange Polynomial through these data points.