Section ActBy.
Variables (aT : finGroupType) (D : {set aT}) (rT : finType).
Definition actby_cond (A : {set aT}) R (to : action D rT) : Prop :=
[acts A, on R | to].
Definition actby A R to of actby_cond A R to :=
fun x a => if (x \in R) && (a \in A) then to x a else x.
Variables (A : {group aT}) (R : {set rT}) (to : action D rT).
Hypothesis nRA : actby_cond A R to.
Canonical action_by := Action actby_is_action.
Local Notation "<[nRA]>" := action_by : action_scope.

Lemma acts_actby (B : {set aT}) S :
[acts B, on S | <[nRA]>] = (B \subset A) && [acts B, on R :&: S | to].