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

theorem Th21:
  c1 * (-c2) = -(c1*c2)
proof
  consider x1,y1,w1,z1 be Element of REAL such that
A1: c1 = [*x1,y1,w1,z1*] by Lm1;
  consider x2,y2,w2,z2 be Element of REAL such that
A2: c2 = [*x2,y2,w2,z2*] by Lm1;
A3: c1 * (-c2) =[*x1,y1,w1,z1*] * [*-x2,-y2,-w2,-z2*] by A1,A2,Th4
    .=[* x1*(-x2)-y1*(-y2)-w1*(-w2)-z1*(-z2),
  x1*(-y2)+y1*(-x2)+w1*(-z2)-z1*(-w2),
  x1*(-w2)+(-x2)*w1+(-y2)*z1-(-z2)*y1,
  x1*(-z2)+z1*(-x2)+y1*(-w2)-w1*(-y2) *] by QUATERNI:def 10
    .=[* -x1*x2+y1*y2+w1*w2+z1*z2,
  -x1*y2-y1*x2-w1*z2+z1*w2, -x1*w2-x2*w1-y2*z1+z2*y1,
  -x1*z2-z1*x2-y1*w2+w1*y2 *];
  c1*c2=[* x1*x2-y1*y2-w1*w2-z1*z2, x1*y2+y1*x2+w1*z2-z1*w2,
  x1*w2+x2*w1+y2*z1-z2*y1,
  x1*z2+z1*x2+y1*w2-w1*y2 *] by A1,A2,QUATERNI:def 10;
  then c1 * (-c2) + (c1*c2)
  =[* -x1*x2+y1*y2+w1*w2+z1*z2+(x1*x2-y1*y2-w1*w2-z1*z2),
  -x1*y2-y1*x2-w1*z2+z1*w2+(x1*y2+y1*x2+w1*z2-z1*w2),
  -x1*w2-x2*w1-y2*z1+z2*y1+(x1*w2+x2*w1+y2*z1-z2*y1),
  -x1*z2-z1*x2-y1*w2+w1*y2+(x1*z2+z1*x2+y1*w2-w1*y2) *]
  by A3,QUATERNI:def 7
    .=[* In(0,REAL),In(0,REAL) *] by QUATERNI:91
    .=0 by ARYTM_0:def 5;
  hence thesis by QUATERNI:def 8;
end;
