SDE-almost sure

Question

Let $(B_t)_{t\geq 0}$ be a Brownian motion. And let $s\leq u\leq t$ Does it then hold that almost surely $$\int_s^uB_rdB_r+\int_u^tB_rdB_r=\int_s^tB_rdB_r?$$ That is for a.e. $\omega\in\Omega$ $$\bigg(\int_s^uB_rdB_r\bigg)(\omega)+\bigg(\int_u^tB_rdB_r\bigg)(\omega)=\bigg(\int_s^tB_rdB_r\bigg)(\omega)?$$ The question why I am asking this is because this property holds in $L^2(\Omega)$ but does it hold almost surely?