A well known result from algebra is that, on any field, there is exactly one polynomial of degree less than n interpolating n points [1, §7].
This entry contains a formalization of the above result, as well as the following generalization in the case of finite fields F: There are |F|m-n polynomials of degree less than m ≥ n interpolating the same n points, where |F| denotes the size of the domain of the field. To establish the result the entry also includes a formalization of Lagrange interpolation, which might be of independent interest.
The formalized results are defined on the algebraic structures from HOL-Algebra, which are distinct from the type-class based structures defined in HOL. Note that there is an existing formalization for polynomial interpolation and, in particular, Lagrange interpolation by Thiemann and Yamada  on the type-class based structures in HOL.