Lemma 2

Let X be a manifold of class C^r, e a point of X, U and V open neighbourhoods of e and m a mapping of class C^r of U × U into X satisfying the following conditions: (a) m(e, x) = m(x, e) = x for all x ∈ U; (b) V ⊂ U, m(V × V) ⊂ U and m(m(x, y), z) = m(x, m(y, z)) for all x, y, z in V.

Then there exists an open neighbourhood W of e in V and an automorphism θ of the manifold W such that θ(e) = e, θ(θ(x)) = x and m(x, θ(x)) = m(θ(x), x) = e for all x ∈ W.

m(e, y) = y for all y ∈ U and hence, by the implicit function theorem, there exists an open neighbourhood W_1 of e in V and a mapping θ_1 of class C^r of W_1 into V such that θ_1(e) = e, m(x, θ_1(x)) = e for all x ∈ W_1. Similarly, there exists an open neighbourhood W_2 of e in V and a mapping θ_2 of class C^r of W_2 into V such that θ_2(e) = e, m(θ_2(x), x) = e for all x ∈ W_2. For x ∈ W_1 ∩ W_2,

Let θ(x) be the common value of θ_1(x) and θ_2(x) for x ∈ W_1 ∩ W_2. Let W be the set of x ∈ W_1 ∩ W_2 such that θ(x) ∈ W_1 ∩ W_2. The set W is open. For x ∈ W,

and hence θ(x) ∈ W. We see that θ | W defines an automorphism of the manifold W.