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

theorem Th18:
  (c1 + c2) * c3 = c1*c3 + c2*c3
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;
  consider x3,y3,w3,z3 be Element of REAL such that
A3: c3 = [*x3,y3,w3,z3*] by Lm1;
A4: (c1+c2)*c3 =[*x1+x2,y1+y2,w1+w2,z1+z2*]*c3 by A1,A2,QUATERNI:def 7
    .=[* (x1+x2)*x3-(y1+y2)*y3-(w1+w2)*w3-(z1+z2)*z3,
  (x1+x2)*y3+(y1+y2)*x3+(w1+w2)*z3-(z1+z2)*w3,
  (x1+x2)*w3+x3*(w1+w2)+y3*(z1+z2)-z3*(y1+y2),
  (x1+x2)*z3+(z1+z2)*x3+(y1+y2)*w3-(w1+w2)*y3 *] by A3,QUATERNI:def 10
    .=[* (x1*x3+x2*x3)-(y1*y3+y2*y3)-(w1*w3+w2*w3)-(z1*z3+z2*z3),
  (x1*y3+x2*y3)+(y1*x3+y2*x3)+(w1*z3+w2*z3)-(z1*w3+z2*w3),
  (x1*w3+x2*w3)+(x3*w1+x3*w2)+(y3*z1+y3*z2)-(z3*y1+z3*y2),
  (x1*z3+x2*z3)+(z1*x3+z2*x3)+(y1*w3+y2*w3)-(w1*y3+w2*y3) *];
  c1*c3+c2*c3 =[* x1*x3-y1*y3-w1*w3-z1*z3, x1*y3+y1*x3+w1*z3-z1*w3,
  x1*w3+x3*w1+y3*z1-z3*y1,
  x1*z3+z1*x3+y1*w3-w1*y3 *]+c2*c3 by A1,A3,QUATERNI:def 10
    .=[* x1*x3-y1*y3-w1*w3-z1*z3,
  x1*y3+y1*x3+w1*z3-z1*w3, x1*w3+x3*w1+y3*z1-z3*y1,
  x1*z3+z1*x3+y1*w3-w1*y3 *]+
  [* x2*x3-y2*y3-w2*w3-z2*z3, x2*y3+y2*x3+w2*z3-z2*w3,
  x2*w3+x3*w2+y3*z2-z3*y2,
  x2*z3+z2*x3+y2*w3-w2*y3 *] by A2,A3,QUATERNI:def 10
    .=[* x1*x3-y1*y3-w1*w3-z1*z3+(x2*x3-y2*y3-w2*w3-z2*z3),
  x1*y3+y1*x3+w1*z3-z1*w3+(x2*y3+y2*x3+w2*z3-z2*w3),
  x1*w3+x3*w1+y3*z1-z3*y1+(x2*w3+x3*w2+y3*z2-z3*y2),
  x1*z3+z1*x3+y1*w3-w1*y3+(x2*z3+z2*x3+y2*w3-w2*y3) *]
  by QUATERNI:def 7;
  hence thesis by A4;
end;
