  This paper is devoted to prove that if an Alexandrov space of curvature not
less than $\kappa$ with a codimension one extremal subset which admits an
isometric involution with respect to the induced length metric, then the metric
space obtained by gluing the extremal subset along the isometry is an
Alexandrov space of curvature not less than $\kappa$. This is a generalization
of Perelman's doubling and Petrunin's gluing theorems.
