reserve q,r,c,c1,c2,c3 for Quaternion;
reserve x1,x2,x3,x4,y1,y2,y3,y4 for Real;

theorem
  for r being Quaternion holds 0q*r = 0
proof
  let r be Quaternion;
  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 thesis by A2,A3,A4,A5,Th11;
end;
