I have been reading up on the below thread: conditional expected value of a brownian motion
but I cannot understand how $$\mathbb E\left[B_s - \frac{s}{t} B_t + \frac{s}{t} B_t\ \mid B_t\right] = \mathbb E\left[B_s - \frac{s}{t} B_t \mid B_t\right] - \mathbb E\left[\frac{s}{t} B_t \mid B_t\right].$$
Is it not supposed to be $\mathbb E\left[B_s - \frac{s}{t} B_t + \frac{s}{t} B_t\ \mid B_t\right]=\mathbb E\left[B_s - \frac{s}{t} B_t \mid B_t\right]+\mathbb E\left[\frac{s}{t} B_t\ \mid B_t\right]$? i.e. a $+$ sign and not $-$ for the last when breaking it out separately.
Happy for all the help I can get!