  We give a construction allowing to lift spectral triples to crossed products
by Hilbert bimodules. The spectral triple one obtains is a concrete unbounded
representative of the Kasparov product of the spectral triple and the
Pimsner-Toeplitz extension associated to the crossed product by the Hilbert
bimodule. To prove that the lifted spectral triple is the above-mentioned
Kasparov product, we rely on operator-$*$-algebras and connexions.
