  We show that for all n-mode Gaussian states of continuous variable systems,
the entanglement shared among n parties exhibits the fundamental monogamy
property. The monogamy inequality is proven by introducing the Gaussian tangle,
an entanglement monotone under Gaussian local operations and classical
communication, which is defined in terms of the squared negativity in complete
analogy with the case of n-qubit systems. Our results elucidate the structure
of quantum correlations in many-body harmonic lattice systems.
