2.1. FIRST RESULTS89but by the associative law, the first two and the last two are equal. Thusthere are at most three different product of four elements:a.bcd/;.ab/.cd/;.abc/d:Using the associative law, we see that all three are equal:a.bcd/Da.b.cd//D.ab/.cd/D..ab/c/dD.abc/d:Thus there is a well-defined product of four elements, which is independentof the way the elements are grouped for multiplication.There are 14 ways to group five elements for multiplication; we won’tbother to list them. Because there is a well-defined product of four or lesselements, independent of the way the elements are grouped for multiplica-tion, there are at most four distinct products of five elements:a.bcde/;.ab/.cde/.abc/.de/;.abcd/e:Using the associative law, we can show that all four products are equal,a.bcde/Da.b.cde//D.ab/.cde/;etc. Thus the product of five elements at a time is well-defined, and inde-pendent of the way that the elements are grouped for multiplication.
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