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We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quadequations on decorated cubes.doi:10.1080/14029251.2013.865829 fatcat:ayva6as2hfgspnl6k22u7vnrb4