How are security proofs performed for Certificateless PEKS schemes?

I have been studying several CLPEKS (CertificateLess Public-key Encryption scheme with Keyword Search) schemes and I understand the security assumptions and the several types of security models used for the security game between the Challenger and the Adversary.

The paper I’m reading is titled "Certificateless searchable public key authenticated encryption with designated tester for cloud-assisted medical Internet of Things"

Below you can see picture of the lemma I’m trying to understand. How do they write the probability to solve the CBDH problem?

My question is what is the concept behind that security proof? What is their aim and what are the rules that guide the process?

I need to understand that to apply it to the scheme I am working on.