Conjugate-join-closedness is left residual-preserved

This article gives the statement, and possibly proof, of a subgroup metaproperty (i.e., Conjugate-join-closed subgroup property (?)) satisfying a subgroup metametaproperty (i.e., Left residual-preserved subgroup metaproperty (?))
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Statement

Property-theoretic statement

The Left residual operator for composition (?) of a conjugate-join-closed subgroup property by any subgroup property is again conjugate-join-closed.