Let Omega subset of R-N be a Lipschitz domain and Gamma be a relatively open and non-empty subset of its boundary partial derivative Omega. We show that the solution to the linear first-order system:del zeta = G zeta, zeta vertical bar Gamma = 0, (1)vanishes if G is an element of L-1 (Omega; R-(NxN)xN) and zeta is an element of W-1,W-1(Omega; R-N). In particular, square-integrable solutions zeta of (1) with G is an element of L-1 boolean AND L-2 (Omega; R((NxN)xN)) vanish. As a consequence, we prove that:.vertical bar vertical bar vertical bar . vertical bar vertical bar vertical bar : C-o(infinity) (Omega, Gamma; R-3) -> [0, infinity), u -> parallel to sym(del uP(-1))parallel to(L2(Omega))is a norm if P is an element of L-infinity (Omega; R-2; R-3x3) with Curl P-1 a L-q (Omega; R-3x3), Curl P-1 E Lq(S2;11t3x3) for some p,q > 1 with 1/p + 1/q = 1 as well as det P >= c(+) > 0. We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let phi is an element of H-1 (Omega; R-3), Omega c R-3, satisfy sym(del Phi(T)del psi) = 0 for some Psi is an element of W-1,W-infinity (Omega;R-3) boolean AND H-2(Omega; R-3) with det del Psi >= c(+) > 0. Then there exists a constant translation vector a is an element of R-3 and a constant skew-symmetric matrix A is an element of so(3), such that Phi = A Psi +a. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.