One of the most studied behavioural equivalences is bisimilarity, andthe main reason for its success is the associated bisimulation proofmethod, which can be further enhanced by means of `up-to bisimulation'techniques such as `up-to context'.A different proof method is discussed, based on unique solutions ofspecial forms of inequations called contractions, and inspired byMilner's theorem on unique-solution of equations. The method is atleast as powerful as the bisimulation proof method and its `up-tocontext' enhancements. The definition of contraction can betransferred onto other behavioural equivalences, possibly contextualand non-coinductive. This enables a coinductive reasoning style onsuch equivalences, either by applying the method based onunique-solution of contractions, or by injecting appropriatecontraction preoders into the bisimulation game.The techniques are illustrated on CCS-like languages; an example withhigher-order languages is also shown.