A long long proof

Decomposition of a relation into join of projections serves as motivation for database normalization theory. In relational lattice terms relation x projected into sets of attributes (that is empty relations) s and t:

x = (x v s) ^ (x v t)

Lets investigate dual perspective and switch the roles of join and inner union:

x = (x ^ s) v (x ^ t)

One particular instance of this identity is known as fundamental decomposition identity

x = (x ^ R00) v (x ^ R11)

which informally asserts that a relation is an inner union of the relation header (i.e. set of attributes) and content (set of tuples). Fundamental decomposition identity can be generalized into

x = (x ^ y`) v (x ^ y')