Proofs in mathematics never claim to be absolutely correct. In fact many mathematical theories contain seemingly contradicting axioms, example:
Euclidean geometry, parralel postulate: Given Line L, Point P, there exists exactly 1 line that passes through P parralel to L
Hyperbolic geometry, parralel postulate: Given Line L, Point P, there exists infinitely many lines that pass through P parralel to L

Those statements do not contradict each other because they are never true at the same time. You are either working in the framework of the euclidean geometry or hyperbolic, or some other geometry. There is no absolute truth in mathematics. (rather, any theorem should be read as "if axioms: ... are true, then theorem: ... is true")

Mathematics does not shows truths, it shows consequences.

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