  We prove holomorphy E sqcap C(I,varPi) to C(I,varPi) of the map (x,y) mapsto
x circ [id,y] where [id,y]:I owns t mapsto (t,y(t)) for a real compact interval
I, and where varPi is a complex Banach space and E is a certain locally convex
space of continuous functions x:I times varPi to varPi for which x(t,.) is
holomorphic for all t in I. We also discuss application of this result to
establishing a holomorphic solution map (xi,varphi) mapsto y for functions y:I
to varPi satisfying the ordinary differential equation y' = varphi circ [id,y]
with initial condition y(t_0) = xi .
