Yes, that's what I meant. But when I peel back the white sheathing, won't
I
just get 8 (or 10) coloured bits which are on their way upstairs? How do
I
get them connected to a new socket AND let them continue their journey
upstairs?

Even if the tool has a cutter, it is usually possible to turn it off. You
can then connect up without cutting the wires, which will probably be good
if there is insufficient slack.

Christian.