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F ( x - y)]* = G(T’ - y’), + + + The independence of the three elements x’,y’,z’ is again a consequence of the independence of the three elements z,y,z. Thus we may apply (5) both on x,y,z and on x’,y’,z’. E - z)l* (FY)*) = [ G(z’ - y’) Gz’] n [ G ( d - z’) Gy’] = G(x’ - y’ - z’) ; and using this result we find similarly that + + + + + [F(y + + z)]* = ([Fy + Fz] n [ F ( x- y - z ) + F 4 ) * = [(FY)* + ( W * I n [ ( F ( z- y z))* + (W*I + Gz’] n [ G(x’ - + + y’ - z’) Gx’] = G(y’ 2’). But the three equations (Fx)* = Gx’, [ F ( x - y - z)]* = G(x’ - Y‘- z’), [ F ( y + z)]* = G(y’ z’) are the three defining equations of h(x,x‘,y Z ) which is uniquely determined by (1); and thus we have h(z,z’,y z ) =y’ Z’ which proves our claim in this first case.

2]. APPENDIX I11 Fano’s Postulate In later parts of our investigation we shall have occasion to exclude the case where the field F [of coordinates] has characteristic 2 [+ 1 = - 11. A geometrical characterization of this fact may be obtained as follows. The four points A , B, C, D in the linear manifold M are said t o form a quadrangle, if they are coplanar [so that A B C D has rank 31 whereas no three of them are collinear [so that A B C=A B D =A C D= B C 4- D + + + + + + + + + + has rank 31. 2 that the lines A B and C D meet in a C and B D point El, the lines A Fig.

H ~k7 of , n elements in W were dependent which contradicts (b). Thus different (n - 1)-tuplets in W span different subspaces in A ; and this proves (2). (3) If S is a subspace of rank i, and if 1 < i, then S contains a t least d subspaces of rank i - 1. Denote by s1,. , s1 a basis of S. If z is an element in F , then let 1-2 s, = Fs, + F(s, 1 4-zsz). , = 1 I t is easy to see that r(S,) = i - 1, and that S, = S, if, and only if, z = y (since the elements s, are independent). Thus we have constructed exactly d distinct subspaces of rank i - 1.