Since X is a finite set and set Orbit is a subset of X which increases strictly at each pass of Step 3 with nonempty New, termination is guaranteed.

By construction of Orbit, it is contained in the G-orbit of x.

If y Gx, then there are b₁, ..., b_r B such that y = b_r ··· b₁(x). By construction, Orbit is invariant under each of the elements in B, so also b_r ··· b₁(x) belongs to Orbit. In particular, y is an element of the output Orbit.