Complex sentence: The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.
Split into simpler sentences: The proof of a mathematical theorem is a logical argument demonstrating that the conclusion is a necessary consequence of the hypotheses. Given such an interpretation of proof as justification of truth, the conclusion is viewed as true in the case that the hypotheses are true, without any further assumptions.