Before revision:  We prove the fact that , if no-arbitrage of the first kind holds on S in F,  the process S ^{\mathfrak{t}-} also has the property of no-arbitrage of the first kind in \mathbb{G}.
After revision:  We prove the fact that , if no-arbitrage of the first kind holds on S in F, the process S ^{\mathfrak{t}-} and S^\mathfrak{t in \mathbb{G}.