ex x1, x2, x3, x4, y1, y2, y3, y4 being Real st
( z = [*x1,x2,x3,x4*] & z9 = [*y1,y2,y3,y4*] & z * z9 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) by Def9;
hence z * z9 in QUATERNION ; :: according to QUATERNI:def 2 :: thesis: verum