  We prove that the Dehn invariant of any flexible polyhedron in Euclidean
space of dimension greater than or equal to 3 is constant during the flexion.
In dimensions 3 and 4 this implies that any flexible polyhedron remains
scissors congruent to itself during the flexion. This proves the Strong Bellows
Conjecture posed by Connelly in 1979. It was believed that this conjecture was
disproved by Alexandrov and Connelly in 2009. However, we find an error in
their counterexample. Further, we show that the Dehn invariant of a flexible
polyhedron in either sphere or Lobachevsky space of dimension greater than or
equal to 3 is constant during the flexion if and only if this polyhedron
satisfies the usual Bellows Conjecture, i.e., its volume is constant during
every flexion of it. Using previous results due to the first listed author, we
deduce that the Dehn invariant is constant during the flexion for every bounded
flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible
polyhedron with sufficiently small edge lengths in any space of constant
curvature of dimension greater than or equal to 3.
