**Proposition 11.2.3**: *Let F = F(t ^{a}, t, t^{b}) be a monomial semi-field type flock in PG(3,q) with q = p^{n} for some prime p, a = p^{i}, b = p^{j} with 0 < i < j < n. Let d = gcd(i,j,n). In the dual setting, every three non-collinear points of D_{F} is contained in an affine plane of points of D_{F} of order p^{d}.*

*Pf*: Let K = GF(p^{d}). Let A, B and C be three non-collinear points of D_{F}. By the proof of Proposition 11.2.2 , we can take A = (0,0,0,1), B = (1,1,1,1) and C = (s^{a},s,s^{b},1) with s not in K (otherwise the points are collinear). Consider the set **S** of p^{2d} points consisting of A and the points (t^{a},t,t^{b},1) where t = 1/u, (s-1)/u or (s+v-1)/uv with u and v in K\{0}. Notice that B and C are amongst these points. We define the lines to be the subsets of **S** which are collinear on a line of PG(3,q). We claim that the linear space **S** is an affine plane, necessarily of order p^{d}. First, we note that all the points of **S** lie in the projective plane of PG(3,q) determined by A, B and C. The points of the line BC (in PG(3,q)), other than B have coordinates of the form (s^{a},s,s^{b},1) + (v-1)(1,1,1,1) = (s^{a}+v-1, s+v-1,s^{b}+v-1,v) for v in GF(q). With v restricted to K\{0}, these coordinates may be rewritten as ((s+v-1/v)^{a},(s+v-1/v),(s+v-1/v)^{b},1). The point of intersection, P, of BC with the plane W'=0 has coordinates ((s-1)^{a},s-1,(s-1)^{b},0). The points on the line joining A to ((s+v-1/v)^{a},(s+v-1/v),(s+v-1/v)^{b},1) have coordinates of the form ((s+v-1/v)^{a},(s+v-1/v),(s+v-1/v)^{b},1) + (u-1)(0,0,0,1) = ((s+v-1/v)^{a},(s+v-1/v),(s+v-1/v)^{b},u) for u in GF(q). Restricting u to K\{0}, gives the points of **S** of the form ((s+v-1/uv)^{a},(s+v-1/uv),(s+v-1/uv)^{b},1). Similarly, the points of AB in **S** are of the form (1/u,1/u,1/u,1) = ((1/u)^{a},1/u,(1/u)^{b},1), and the points of AP in **S** are of the form ((s-1/u)^{a},(s-1)/u,(s-1/u)^{b},1). Given two points of **S**, the line joining them in PG(3,q) contains p^{d} points of D_{F} by Proposition 11.2.1 , and by the proof of that proposition , they all lie in **S**. Hence, **S** contains p^{2d} points, every two points lie on a unique line and each line contains exactly p^{d} points. By Dembowski, Theorem 3.2.4(b) [De67] this is sufficient to show that **S** is an affine plane of order p^{d}.