The study presented in this report is part of a wide research project concerning argumentation and proof. Starting from the notion of mathematical theorem as the unity of statement, proof and theory, a structural analysis of proofs by contradiction has been carried out, producing a model to be used in the observation, analysis, and interpretation of cognitive and didactical issues related to this particular type of proof. In particular, the model highlights the complex relationship between the original statement to be proved and a new statement (the secondary statement) that is actually proved. Through the analysis of an exemplar protocol, this paper discusses on cognitive difficulties concerning the relationship between the reference theory and the proof of the secondary statement.