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

theorem Th22:
  (-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;
  -c1 = [*-x1,-y1,-w1,-z1*] by A1,Th4;
  then (-c1) * (-c2)=[*-x1,-y1,-w1,-z1*] * [*-x2,-y2,-w2,-z2*] by 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 by A1,A2,QUATERNI:def 10;
  hence thesis;
end;
