Up to

Two mathematical objects a and b are called equal up to an equivalence relation R
* if a and b are related by R, that is,
* if aRb holds, that is,
* if the equivalence classes of a and b with respect to R are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.For example, x is unique up to R means that all objects x under consideration are in the same equivalence class with respect to the relation R.