Hi, everyone: I am going over J.Lee's Smooth Manifolds, Chapter 3; specifically, Lemmas 3.1, 3.4, in which he states properties of derivations. Lee calls linear maps L with the Leibniz property (i.e L(fg)(a)=f(a)L(g)+g(a)L(f) ) derivations, when these maps are defined in a subset of R^n, and he calls these same maps tangent vectors when these maps are defined on the tangent space of a general (abstract) manifold. So far, so good. **Now** , what I find confusing is this: that the lemmas are cited separately, as if the two lemmas need different proofs; I don't see why different proofs are necessary. The two properties cited (same properties cited in lemmas 3.1, 3.4 respectively; a),b) below are in lemma 3.1, and a'), b') are in lemma 3.4) ) are : (f is in C^oo(M) , a is a point in R^n , p is any point on the manifold M) a) If f is a constant function, then Lf=0 b) If f(a)=g(a) , then L(fg)(a)=0 a') If f is a constant function, then Xf=0 b') If f(p)=g(p)=0 , then X(fg)(p)=0 The proof for a),b) are straightforward : a) L(c)=X(1c)=1X(c)+cX(1)=cX(1)+cX(1) b) L(fg)(p)=f(p)L(g)+g(p)L(f) Now, why do we need separate proofs of the same facts for a') and b').? Thanks.