(6) ⇒ (7) for otherwise the extended plane ℂ ∪ {∞} \ "G" can be written as the disjoint union of two open and closed sets "A" and "B" with ∞ in "B" and "A" bounded. Let δ > 0 be the shortest euclidean distance been "A" and "B" and build a square grid on ℂ with length δ/4 with a point "a" of "A" at the centre of a square. Let "C" be the compact set of the union of all squares with distance ≤ δ/4 from "A". Then "C" ∩ "B" = ∅ and ∂"C" does not meet "A" or "B": it consists of finitely many horizontal and vertical segments in "G" forming a finite number of closed rectangular paths γ"j" in "G" . Taking "C""i" to be all the squares covering "A", the (2 π)−1 ∫∂"C" d arg("z" − "a") equals the sum of the winding numbers of "C""i"