Polynomial/K/Interpolation/Fact

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}n}$ different elements ${\displaystyle {}a_{1},\ldots ,a_{n}\in K}$, and ${\displaystyle {}n}$ elements ${\displaystyle {}b_{1},\ldots ,b_{n}\in K}$ be given.

Then there exists a unique

polynomial ${\displaystyle {}P\in K[X]}$ of degree ${\displaystyle {}\leq n-1}$, such that ${\displaystyle {}P{\left(a_{i}\right)}=b_{i}}$

holds for all ${\displaystyle {}i}$.