"in a different way..." or "in the different way"

[zorank]

I am not sure how to formulate the following text

"The first two corollaries can be proven easily without using the Lemma. In here they are proven in a different way to illustrate..."

or should it be

"The first two corollaries can be proven easily without using the Lemma. In here they are proven in the different way to illustrate..."

?

Regards
Zoran

[BobK]

The first, usually. If the context has specified the new way, it's possible to use 'the': 'On dance floors in the '60s more and more people were dancing in the new way [described in an earlier sentence].

b

[zorank]

Aha! The dance example was illuminating. Thanks!