Conditional expectation equation containing martingales
Actually, the definition of martingale gives $$\tag{*} \mathbb E\left[e^{i\lambda M_t+\frac12\lambda^2t}|\mathcal{F}_s\right]=e^{i\lambda M_s+\frac12\lambda^2s}. $$ We have $$ \mathbb E\left[e^{i\lambda (M_t-M_s)}|\mathcal{F}_s\right]=\mathbb E\left[e^{i\lambda M_t}e^{-i\lambda M_s}|\mathcal{F}_s\right]=e^{-i\lambda M_s}\mathbb E\left[e^{i\lambda M_t}|\mathcal{F}_s\right] $$ and $$ \mathbb E\left[e^{i\lambda M_t}|\mathcal{F}_s\right]=\mathbb E\left[e^{i\lambda M_t+\frac12\lambda^2t}|\mathcal{F}_s\right]e^{-\frac12\lambda^2t} $$ and (*) gives the wanted result.