Cross-lingual induction aims to acquire for one language some linguistic structures resorting to annotations from another language. It works well for simple structured predication problems such as part-of-speech tagging and dependency parsing, but lacks of significant progress for more complicated problems such as constituency parsing and deep semantic parsing, mainly due to the structural non-isomorphism between languages. We propose a decomposed projection strategy for cross-lingual induction, where cross-lingual projection is performed in unit of fundamental decisions of the structured predication. Compared with the structured projection that projects the complete structures, decomposed projection achieves better adaptation of non-isomorphism between languages and efficiently acquires the structured information across languages, thus leading to better performance. For joint cross-lingual induction of constituency and dependency grammars, decomposed cross-lingual induction achieves very significant improvement in both constituency and dependency grammar induction.