Proposition 7

Every connected group G which is locally isomorphic to $\mathbf{R}$ is isomorphic to either $\mathbf{R}$ or $\mathbf{T}$.

For a local isomorphism of $\mathbf{R}$ with G extends to a strict morphism of $\mathbf{R}$ onto an open subgroup of G (Corollary to Proposition 6), hence onto G itself since G is connected. Hence G is isomorphic to a quotient group of $\mathbf{R}$; since G is Hausdorff and does not consist of the identity element alone (because it is locally isomorphic to $\mathbf{R}$), it is isomorphic to either $\mathbf{R}$ or $\mathbf{T}$ by Proposition 3 of no. 2.