Proof: Let and be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then ; since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then ; since p1 = p2, then cp1 = cp2, so cp is an element of W.
proof let and be elements of w that is points in the plane such that p1 = p2 and q1 = q2 then ; since p1 = p2 and q1 = q2 then p1 + q1 = p2 + q2 so p + q is an element of w let p = (p1 p2) be an element of w that is a point in the plane such that p1 = p2 and let c be a scalar in r then ; since p1 = p2 then cp1 = cp2 so cp is an element of w