  Proving for triangulations an extended version of the 4-colour theorem by
induction, we manage to exclude the case which led to the failure of Kempe's
attempted proof. The new idea is to claim the existence of a "nice"
4-colouring, in which existing Kempe chains satisfy a special condition, and to
show that the assumption of its non-existence by a counterexample always leads
to a contradiction: hence there is such a colouring. The focus of this paper is
not to find in the induction step a reducible colouring of the neighbourhood of
a 5-valent vertex among all possible colourings -- which is known to be
impossible - but to prove the mere existence of such a "nice" colouring among
all possible colourings.
