To Prove:  The total cost for each of the two persons can be reduced, without affecting the total quantities produced, by altering the quantities produced by each individual (i. e., by specialization and exchange).

Proof:    Since it is arbitrary which person is called "A" and which is "B," we may assume that in the above inequality, the left side is greater than the right—that is:

(If the reverse were true, then the remainder of this proof could still be applied by switching the A and B labels and the a and b subscripts.) Therefore there exists at least one positive number t such that: