Formula used:

1) Greatest than definition:

Let D be an ordered integral domain with \(D^{+}\) as the set of positive elements. The relation greater than, denoted by >, is defined on elements x and y of D by

\(\displaystyle{x}\ {>}\ {y}\ \text{if and only if}\ {x}\ -\ {y}\ \in{D}^{+}\).

2) Property of greater that (>):

\(\displaystyle\text{If}\ {x}\ {>}\ {y}\ \text{and}\ {z}\ {>}\ {0},\text{then}\ {x}{z}\ {>}\ {y}{z}\)

Proof:

\(\displaystyle\text{Let}\ {\left[{a},{b}\right]}\ {>}\ {\left[{c},{d}\right]}\)

\(\displaystyle\text{Then}\ {\left[{a},{b}\right]}-{\left[{c},{d}\right]}\in\ {Q}^{+}\)

\(\displaystyle\text{So},{\left[{a},{b}\right]}\ +\ {\left[-{c},{d}\right]}\in{Q}^{+}\)

\(\displaystyle\text{Therefore},{\left[{a}{d}\ -\ {b}{c},{b}{d}\right]}\in{Q}^{+}\)

\(\displaystyle\text{This implies that}{\left({a}{d}\ -\ {b}{c}\right)}{b}{d}\in{D}^{+}\)

\(\displaystyle\text{So},{a}{d}{b}^{{2}}\ -\ {c}{d}{b}^{{2}}\in{D}^{+}\)

Conversely,

\(\displaystyle\text{Let}\ {a}{b}{d}^{{2}}\ -\ {c}{d}{b}^{{2}}\in{D}^{+}\)

\(\displaystyle\text{Therefore},{\left({a}{d}\ -\ {b}{c}\right)}{b}{d}\in{D}^{+}\)

\(\displaystyle\text{So},{\left[{a}{d}\ -\ {b}{c},{b}{d}\right]}\in{Q}^{+}\text{as}\ {b}{d}\ne{q}{0}\)

\(\displaystyle\text{Then}{\left[{a},{b}\right]}\ +\ {\left[-{c},{d}\right]}\in{Q}^{+}\)

\(\displaystyle\text{Implies that}{\left[{a},{b}\right]}\ -\ {\left[{c},{d}\right]}\in{Q}^{+}\)

\(\displaystyle\text{Hence},{\left[{a},{b}\right]}\ {>}\ {\left[{c},{d}\right]}\)

\(\displaystyle\text{Therefore},{\left[{a},{b}\right]}\ {>}\ {\left[{c},{d}\right]}\text{if and only if}\ {a}{b}{d}^{{2}}\ -\ {c}{d}{b}^{{2}}\in{D}^{+}\)