Basic theorems in algebraic K-theory

In mathematics, there are several theorems basic to algebraic "K"-theory.
Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.)
Theorems.
The localization theorem generalizes the localization theorem for abelian categories.
Let formula_1 be exact categories. Then "C" is said to be cofinal in "D" if (i) it is closed under extension in "D" and if (ii) for each object "M" in "D" there is an "N" in "D" such that formula_2 is in "C". The prototypical example is when "C" is the category of free modules and "D" is the category of projective modules.