let r be Quaternion; 0q * r = 0
consider x1, x2, x3, x4, y1, y2, y3, y4 being Real such that
A1:
0 = [*x1,x2,x3,x4*]
and
r = [*y1,y2,y3,y4*]
and
A2:
0q * r = [*((((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 QUATERNI:def 10;
A3:
x1 = 0
by A1, Th11, QUATERNI:12;
A4:
x2 = 0
by A1, Th11, QUATERNI:12;
A5:
x3 = 0
by A1, Th11, QUATERNI:12;
x4 = 0
by A1, Th11, QUATERNI:12;
hence
0q * r = 0
by A2, A3, A4, A5, Th11; verum