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We express a general multiple polylogarithm of weight n as an explicit linearcombination of multiple polylogarithms of weight n in n-2 variables. We expressa general multiple polylogarithm of weight 4 as an explicit linear combinationof multiple polylogarithms of type (3,1). We deduce a 4 parameters functionalequation expressing a certain linear combination of multiple polylogarithms oftype (3,1) as a linear combination of polylogarithms of weight 4.

Sur la conjecture de Zagier pour n=4. II