End DirectSums.
Section CardGL.
Variable F : finFieldType.

(* An alternate, somewhat more elementary proof, that does not rely on the *)
(* row-space theory, but directly performs the LUP decomposition.          *)
Lemma LUP_card_GL n : n > 0 ->
#|'GL_n[F]| = (#|F| ^ 'C(n, 2) * \prod_(1 <= i < n.+1) (#|F| ^ i - 1))%N.