Corollary 2

The map $(s, t) \mapsto C(s, t)$ of $J \times J$ into $\mathscr{L}(E)$ is continuous.

By (6) we have $C(s, t) = C(s, t_0) \left( C(t, t_0) \right)^{-1}$; now, the map $(X, Y) \mapsto XY$ of $\mathscr{L}(E) \times \mathscr{L}(E)$ into $\mathscr{L}(E)$ is continuous, as is the map $X \mapsto X^{-1}$ of the (open) group of invertible elements of $\mathscr{L}(E)$ onto itself (TG, IX, p. 40, prop. 14).

One may note that the map

admits a derivative equal to $-(C(t, t_0))^{-1} (dC(t, t_0)/dt)(C(t, t_0))^{-1}$ (on the complement of a countable set) (I, p. 8, prop. 4), that is to say (by IV, p. 179, formula (5)) equal to $-C(t_0, t) A(t)$.