Finite-join-closedness is left residual-preserved

Property-theoretic statement
The fact about::left residual operator for composition of a finite-join-closed subgroup property by any subgroup property is again finite-join-closed.

Similar facts about being left residual-preserved

 * Join-closedness is left residual-preserved
 * Finite-intersection-closedness is left residual-preserved
 * Intersection-closedness is left residual-preserved
 * Conjugate-join-closedness is left residual-preserved
 * Normalizing join-closedness is left residual-preserved

Similar facts about being right residual-preserved
Some similar facts about right residual-preserved are:


 * Upper join-closedness is right residual-preserved
 * Intermediate subgroup condition is right residual-preserved