International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 54

Let be factored into a product of pairwise coprime integers, so that g.c.d. for . Then the system of congruence equationshas a unique solution mod N. In other words, each is associated in a onetoone fashion to the dtuple of its residue classes in .
The proof of the CRT goes as follows. LetSince g.c.d. there exist integers and such thatthen the integeris the solution. Indeed,because all terms with contain as a factor; andby the defining relation for .
It may be noted thatso that the are mutually orthogonal idempotents in the ring , with properties formally similar to those of mutually orthogonal projectors onto subspaces in linear algebra. The analogy is exact, since by virtue of the CRT the ring may be considered as the direct productvia the two mutually inverse mappings:
The mapping defined by (ii) is sometimes called the `CRT reconstruction' of from the .
These two mappings have the property of sending sums to sums and products to products, i.e:(the last proof requires using the properties of the idempotents ). This may be described formally by stating that the CRT establishes a ring isomorphism: