Proof: A proof of a statement is a valid argument to the effect that the statement follows from given assumptions. If these assumptions are true, the statement proved to follow from them is also true.
There are many styles of proof, such as a rhetorical proof in a debate or an argument in law, but the best proofs are those which can be cast in a logical or mathematical format and then still hold - which often means that quite a few things have to be explicated or explicitly assumed that are left tacit or are indeed missed in other styles of proof.
In logic, a valid proof has the property used above: If the assumptions used in the proof are true, then the conclusions deduced from them are also true, for this follows from the definition of valid.
It should be noted that even in mathematics most proofs are informal, at least in the sense that in most proofs much detail is left out that is taken as self-evident, at least for those mathematicians who are familiar with the subject. This also has a good reason: It would be boring and would take much more space.
See also: Argument, Deduction, Evidence, Fallacy, Logic, Logical Terms, Mathematical Logic, Natural Deduction, Proof Theory, Propositionna Logic, Rational, Wisdom, Wishful Thinking,
Cartwright, Hilbert & Bernays, Wang