  Starting from an extension of the Poisson bracket structure and
Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous
systems to spin systems, defined on a lattice, we derive a series of, as we
think, new and interesting bounds on correlation functions for general lattice
systems. Our method is expected to yield also useful results in Euclidean Field
Theory. Furthermore the approach is applicable in situations where other
techniques fail, e.g. in the study of phase transitions without breaking of a
{\bf continuous} symmetry like $P(\phi)$-theories with $\phi (x)$ scalar.
