Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P that contains 1. Prove that N is a normal subgroup of G and that P is the set of its cosets.
This exercise was asked about in the following questions: partition of a group to have normal subgroup partition of a group and cosets Proving a partition is set of cosets Artin 2.10.3 help understanding why $AN=NA=A$ implies $N$ is normal
My question is about Brian Bi's proof linked here, where it is claimed that $1 \in P_n$.
The following is a screenshot of the proof (Kiefer Sutherland's voice):
Please explain the $1 \in P_n$. This is the only part I don't understand.