Theorem:For every element of can be expressed as a product of transpositions.

Thus, given any premutation we can write it as a product of transpositions and we will show that each of these transpositions is a product of the form .

Say we are working in (products are taken from right to left).

Consider . We can write it as .

Consider . We can write it as .

Consider . We can write it as .

You get the general idea. So that means everything can be expressed in consecutive form by using the theorem.