Kernel of a multihomomorphism implies completely divisibility-closed

Statement

Suppose $G$ and $M$ are groups, $n\ge 2$ and:

$b:G\times G\times G\times \dots G\to M$

(where the $G$ occurs $n$ times) is a multihomomorphism. Define:

$P:=\{x\in G\mid b(x,x_2,x_3,\dots ,x_n)\text{ is the identity element of }M\text{ for all }{x}_2,x_3,\dots ,x_n\in G\}$

Then, $P$ is a completely divisibility-closed subgroup of $G$.

Facts used

1. Kernel of a multihomomorphism implies intersection of kernels of bihomomorphisms
2. Intersection of kernels of bihomomorphisms implies completely divisibility-closed

Proof

The proof follows by combining Facts (1) and (2).