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

theorem Th17:
  c1 * (c2 + c3) = c1*c2 + c1*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) =c1*[*x2+x3,y2+y3,w2+w3,z2+z3*] by A2,A3,QUATERNI:def 7
    .=[* x1*(x2+x3)-y1*(y2+y3)-w1*(w2+w3)-z1*(z2+z3),
  x1*(y2+y3)+y1*(x2+x3)+w1*(z2+z3)-z1*(w2+w3),
  x1*(w2+w3)+(x2+x3)*w1+(y2+y3)*z1-(z2+z3)*y1,
  x1*(z2+z3)+z1*(x2+x3)+y1*(w2+w3)-w1*(y2+y3) *] by A1,QUATERNI:def 10
    .=[* (x1*x2+x1*x3)-(y1*y2+y1*y3)-(w1*w2+w1*w3)-(z1*z2+z1*z3),
  (x1*y2+x1*y3)+(y1*x2+y1*x3)+(w1*z2+w1*z3)-(z1*w2+z1*w3),
  (x1*w2+x1*w3)+(x2*w1+x3*w1)+(y2*z1+y3*z1)-(z2*y1+z3*y1),
  (x1*z2+x1*z3)+(z1*x2+z1*x3)+(y1*w2+y1*w3)-(w1*y2+w1*y3) *];
  c1*c2+c1*c3 =[* 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*c3 by A1,A2,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 *]+
  [* 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 *] by A1,A3,QUATERNI:def 10
    .=[* x1*x2-y1*y2-w1*w2-z1*z2+(x1*x3-y1*y3-w1*w3-z1*z3),
  x1*y2+y1*x2+w1*z2-z1*w2+(x1*y3+y1*x3+w1*z3-z1*w3),
  x1*w2+x2*w1+y2*z1-z2*y1+(x1*w3+x3*w1+y3*z1-z3*y1),
  x1*z2+z1*x2+y1*w2-w1*y2+(x1*z3+z1*x3+y1*w3-w1*y3) *]
  by QUATERNI:def 7;
  hence thesis by A4;
end;
