Suppose E,F, G are as in the fundamental theorem and E is generated by two intermediate extensions K and L such that K n L = F and L/F is normal. Let N = Gal E/L, H = Gal E/K. Show that N is normal in G, H n N = 1 and G = HN, so G is the semi-direct product of N and H. Show also that if K/F and L/F are normal then G = H x N.