Dynamic Epistemic Logic (DEL) deals with the representation of situations in a multi-agent and dynamic setting. It can express in a uniform way statements about: (i) what is true about an initial situation (ii) what is true about an event occurring in this situation (iii) what is true about the resulting situation after the event has occurred. After proving that what we can infer about (ii) given (i) and (iii) and what we can infer about (i) given (ii) and (iii) are both reducible to what we can infer about (iii) given (i) and (ii), we provide a tableau method deciding whether such an inference is valid. We implement it in LOTRECscheme and show that this decision problem is NEXPTIME-complete. This contributes to the proof theory and the study of the computational complexity of DEL which have rather been neglected so far.