I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e.,
$$B_{1t} - B_{2t} = \sqrt{2}B_t$$
where $B_{1t}$ and $B_{2t}$ are independent brownian bridge processes, but I can't seem to see why that is. Can someone validate that it's a true statement? Or is the difference of two brownian bridges something else?