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:

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

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

Facts used

 * 1) uses::Kernel of a multihomomorphism implies intersection of kernels of bihomomorphisms
 * 2) uses::Intersection of kernels of bihomomorphisms implies completely divisibility-closed

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