Combining the first variation formula together with the fact that the variations δ δ λ λ u σ σ are symmetries, we get 0 = E σ σ δ δ λ λ u σ σ + d i W λ λ i , W λ λ i = ∑ ∑ | I | = 0 r P σ σ i I δ δ λ λ u σ σ , where on the first term proportional to the Euler-Lagrange expressions, further integrations by parts can be performed as E σ σ δ δ λ λ u σ σ = ∑ ∑ | I | = 0 s E σ σ R a σ σ , I λ λ I a = Q a λ λ a + d i ( ∑ ∑ | I | = 0 s − − 1 Q a i I λ λ I a ) , where Q a I = ∑ ∑ | J | = 0 s − − | I | ( − − 1 ) | J | d J ( E σ σ R a σ σ , I J ) , in particular for | I | = 0 , Q a = E σ σ R a σ σ − − d i ( E σ σ R a σ σ , i ) + ⋯ ⋯ + ( − − 1 ) s d i 1 … … d i s ( E σ σ R a σ σ , i 1 . . . i s ) = ∑ ∑ | I | = 0 s ( − − 1 ) | I | d I ( E σ σ R a σ σ , I ) . Hence, we have an off-shell relation 0 = Q a λ λ a + d i S λ λ i , where S λ λ i = H λ λ i + W λ λ i , with H λ λ i = ∑ ∑ | I | = 0 s − − 1 Q a i I λ λ I a . This relation is valid for any choice of the gauge parameters λ λ a ( x ) . Choosing them to be compactly supported, and integrating the relation over the manifold of independent variables, the integral total divergence terms vanishes due to Stokes' theorem. Then from the fundamental lemma of the calculus of variations, we obtain that Q a ≡ ≡ 0 identically as off-shell relations (in fact, since the Q a are linear in the Euler-Lagrange expressions, they necessarily vanish on-shell). Inserting this back into the initial equation, we also obtain the off-shell conservation law d i S λ λ i = 0 .