Corollary of Krull-Remak-Schmidt theorem for cancellation of direct factors

Statement
Suppose $$G,H,K$$ are groups such that both $$G$$ and $$H$$ satisfy the ascending chain condition on normal subgroups and descending chain condition on normal subgroups. Then, the external direct product $$G \times H$$ is isomorphic to the external direct product $$G \times K$$, we must also have that $$H$$ and $$K$$ are isomorphic groups.

Weaker facts

 * Direct product is cancellative for finite groups

Facts used

 * 1) uses::Krull-Remak-Schmidt theorem