  Lebesgue measurable subsets A and B of parallel or identical k-dimensional
affine subspaces of Euclidean n-space E^n satisfy
  The Product Formula for Volume: Vol_k(A)Vol_k(B) = \sum_{J \in S(n,k)}
Vol_k({\pi}_J(A))Vol_k({\pi}_J(B)).
  Here Vol_k denotes k-dimensional Lebesgue measure; S(n,k) denotes the set of
all k-element subsets of {1,2,..., n}; and for J \in S(n,k), E^J =
{(x_1,x_2,...,x_n) \in E^n : x_i = 0 for all i \notin J} and {\pi}_J : E^n
\rightarrow E^J is the projection that sends the i^{th} coordinate of a point
of E^n to 0 whenever i \notin J.
  Setting B = A, we obtain the corollary:
  The Pythagorean Theorem for Volume: Vol_k(A)^2 = \sum_{J \in S(n,k)}
(Vol_k({\pi}_J(A)))2.
