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The second problem concerns the distance between the machine and the
position of the 'new' socket; I'd need a machine with a cable of around 3m
long to cover the distance.

Next to the existing single socket, install an 13A RCD fused
connection unit. If the entire circuit is protected by a 30mA
immediate action RCD already, a standard fused connection unit will do
(at much lower cost). Run T&E cable from this (i.e. 2.5mm T&E) to a
flex outlet in the bathroom, not near to the bath or shower. Cut the
plug off the drier's flex and connect to the flex outlet. You mustn't
install the drier or flex outlet within 60cm of the bath or shower.
You mustn't install a socket instead of a flex outlet. You mustn't run
the cable near to the bath or shower.

Christian.