An imbedding property of K-groups

For a pair of rings A⊂BA⊂B satisfying a certain condition (⋆)(⋆) we get general imbedding results for Karoubi–Villamayor, homotopy and Quillen K-theories. One important case is when AA and BB are respectively the henselization and completion of a regular ring of finite type over a field at a prime ideal.