Consider the action of the nonzero real numbers R^{x}
on R^{n}
t.e_{j} = t ^{j} e_{j} ,  (4.10) 
0 =  t^{n(n+1)/2b} L_{b} p_{b}(K) , 

(4.11) 
Proof.
We induct on m to construct numbers
t_{1}, t_{2}, ...,
t_{k(nk)} in
R^{x} having the property that, for all
a in C_{n,k}
with a = m, the intersection
of the Schubert varieties
The case m=0 is trivial, as a=0 implies that a=0 and X_{0} is a single (real) point in Gr(k, n). Suppose we have constructed t_{1}, t_{2}, ..., t_{m} in R^{x} with the above properties.
Let a be in C_{n, k} with a=m+1 and set Z_{0} to be the intersection of the Schubert variety X_{a} with the hyperplane defined by p_{a} = 0. By Theorem 4.5, Z_{0} is the union of the Schubert varieties X_{b} for all b in C_{n,k} that are covered by a. Consider the intersection of Z_{0} with the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L). By our assumption that each Schubert variety X_{b} with b=m meets the intersection of the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L) transversally, Z_{0} will meet that intersection of Schubert varieties transversally if no two Schubert varieties X_{b} and X_{c} with b=c=m have a point in common with that intersection.
Since the intersection of X_{b} and X_{c} is a union of Schubert varieties (of smaller dimension), this would imply that a Schubert variety X_{e} of dimension less than m meets the intersection of the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L). Consider a fixed kplane K in X_{e}. By Equation (4.11), the condition for t.L to meet K nontrivially (that is ,for K to lie in X(t.L)) is a polynomial in t of degree at most e, which is less than m. Thus no kplane K in some Schubert variety X_{e} with dimension less than m lies in the intersection of the m Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L). We conclude that the intersection of Z_{0} with those Schubert varieties is transverse.
Since each component X_{b} of Z_{0} meets that intersection of Schubert varieties in d(b) real points, we conclude that Z_{0} meets that intersection transversally in the sum over all b covered by a of d(b) real points. By (4.8) this sum is just d(a).
For t in R, let Z_{t} be set of points in X_{a} that satisfy the polynomial (4.11). For nonzero t, this is the intersection of X_{a} with the Schubert variety X(t.L), that is, those kplanes in X_{a} that meet t.L nontrivially. Since L has no vanishing Plücker coordinates, the constant term of that polynomial is p_{a}(K), and so by Theorem 4.5, Z_{0} is the intersection of X_{a} with the hyperplane H_{a} defined by p_{a}(K)=0.
By our previous arguments, Z_{0} meets the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L) transversally (over C) in d(a) real points. Thus there exists a small positive number number E_{a} such that for t between 0 and E_{a} (inclusive), Z_{t} meets the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L) transversally (over C) in d(a) real points. Thus, for t in this range, X_{a} meets the Schubert varieties X(t.L), X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L) transversally (over C) in d(a) real points.
Let t_{m+1} be the minimum of these numbers E_{a} over all a with weight a = m+1. Then for any a in C_{n, K} with a = m+1, X_{a} meets the Schubert varieties X(t_{1}.L), X(t_{2}.L), ..., X(t_{m}.L), X(t_{m+1}.L) transversally (over C) in d(a) real points. Q.E.D.