You cannot establish what was already established in advance. To establish a country is to create one that did not exist before. Analogously, to establish a conclusion is presumably also to bring the audience to believe what they did not believe firmly before. However, we often argue for conclusions that everybody already strongly believed in advance. Just imagine that one mathematician had already proven the Pythagorean theorem (the square of the hypotenuse in a right triangle is equal to the sum of the squares of the other two sides). Then another mathematician comes up with a new proof that is shorter and makes fewer assumptions. Both proofs are arguments, but the purpose of proving the theorem the second time is not to convince people who did not believe the theorem. Everyone already believed it. Yet mathematicians still might want to prove it in fewer steps with fewer assumptions in order to determine why it is true and which axioms or premises its truth depends on. Their proof aims to explain the theorem but not to establish it.