  The long standing problem is solved why the number and the location of
monopoles observed in Lattice configurations depend on the choice of the gauge
used to detect them, in contrast to the obvious requirement that monopoles, as
physical objects, must have a gauge-invariant status.
  It is proved, by use of non-abelian Bianchi identities, that monopoles are
indeed gauge-invariant: the technique used to detect them has instead an
efficiency which depends on the choice of the abelian projection, in a known
and controllable way.
