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This paper presents the "K-language" for generalized monadic recursion schemes, and also presents a formal axiom system which derives strong equivalences among monadic recursion schemes. The distinct features of the K-system are (1)that each scheme has two exits, but its control structure is still well-structured. Therefore, this can be a candidate of a new extended framework of control structure in computer languages; and (2)that the equivalence-proving ability of the K-system seems to be the most powerful among all systems proposed before. The axiom system apparently is a kind of a mixture of Boolean algebra and Salomaa's formal system for the regular expression.