ex q0, q1, q2, q3, r0, r1, r2, r3 being Element of REAL st
( q = [*q0,q1,q2,q3*] & r = [*r0,r1,r2,r3*] & q / r = [*(((((r0 * q0) + (r1 * q1)) + (r2 * q2)) + (r3 * q3)) / (|.r.| ^2)),(((((r0 * q1) - (r1 * q0)) - (r2 * q3)) + (r3 * q2)) / (|.r.| ^2)),(((((r0 * q2) + (r1 * q3)) - (r2 * q0)) - (r3 * q1)) / (|.r.| ^2)),(((((r0 * q3) - (r1 * q2)) + (r2 * q1)) - (r3 * q0)) / (|.r.| ^2))*] ) by Def1;
hence q / r is quaternion ; :: thesis: verum