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Proof

Proposition: forall A, B: A union B = B union A

Proof: Take arbitrary A and B. By definition of =, we have to prove

(1) forall x: x in A union B <=> x in B union A.

• We prove x in A union B => x in B union A. Assume

  (2) x in A union B.

  We have to prove x in B union A. By definition of union , we have to prove

  (3) x in B \/ x in A.

  If x in B, we are done. Thus assume (4) x not  in B. By (2) and definition of union , we have (5) x in A \/ x in B. From (4) and (5), we have x in A and thus (3).
• The proof of x in B union A => x in A union B proceeds analogously.