oo Proof. 4 that there exists a common neighborhood G* in which z(x, Y) and zv(x, Y) are defined. 4). 6) are of class Cl in a neighborhood V*of the point x = x H = Y".

4) satisfying (6. 5) and (8. 5). But, Y(x) and Yi(x) being two solutions of (8. 4) passing through the point (x*, Y*) we have by the property PI lim x}x* z (x. , n) . ,n) , (x*,Yi(x*)) z (,,x*, _ x=x* x=x* zl(x*) = z(x''`,Yi(x*)) = z(x*, Y(x*)), r and, in particular, q(x*) i = zy (x*, Yl(x*)) = zy Y(x*)) i J it follows from (8, 5), (22. 5), (23. 5) and (24. 5) that the functions Y(x), z(x,Y(x)), zy(x, Y(x)) satisfy characteristic equations (5. 4). This completes the proof that property P1 implies property P.