Corollary of Timmesfeld's replacement theorem for elementary abelian subgroups

Statement
Suppose $$P$$ is a group of prime power order and $$A$$ is an fact about::elementary abelian subgroup of maximum rank in $$P$$. Suppose $$V$$ is an $$A$$-invariant elementary abelian subgroup of $$P$$. Then, $$[V,A]C_A([V,A])$$ is an elementary abelian subgroup of maximum rank.

Related facts

 * Corollary of Timmesfeld's replacement theorem for abelian subgroups
 * Timmesfeld's replacement theorem
 * Thompson's replacement theorem for abelian subgroups
 * Glauberman's replacement theorem

Facts used

 * 1) uses::Timmesfeld's replacement theorem