How are prime implicates of HORN-Formulas defined?

I’m confused about the definition of prime implicates in Horn formulas.

For example in the paper of Kira 2012 on page 109 it is stated: enter image description here

Now in the paper of Boros 2010 on page 82 the following definition is used: enter image description here

My goal is to decide whether a Horn formula is prime or not in polynomial time. For that I want to assume the definition used in Kira 2012.

How can I prove that the two definitions above for prime implications of Horn formulas are equivalent?