Direct factor implies join-transitively central factor

Statement
Any direct factor of a group is a join-transitively central factor: its join with any central factor is a central factor.

Facts used

 * 1) uses::Direct factor implies right-quotient-transitively central factor
 * 2) uses::Right-quotient-transitively central factor implies join-transitively central factor

Proof
The proof follows by piecing together facts (1) and (2).