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

theorem
  r" = Rea r / (|.r.|^2) - Im1 r / (|.r.|^2)*<i> -
  Im2 r / (|.r.|^2)*<j> - (Im3 r) / (|.r.|^2)*<k>
proof
A1: 1q =[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*jj,0,0,0*] by QUATERNI:91;
  consider q0,q1,q2,q3,r0,r1,r2,r3 being Element of REAL such that
A2: 1q = [*q0,q1,q2,q3*] and
A3: r = [*r0,r1,r2,r3*] and
A4: r" = [* (r0*q0+r1*q1+r2*q2+r3*q3)/(|.r.|^2),
  (r0*q1-r1*q0-r2*q3+r3*q2)/(|.r.|^2),
  (r0*q2+r1*q3-r2*q0-r3*q1)/(|.r.|^2),
  (r0*q3-r1*q2+r2*q1-r3*q0)/(|.r.|^2) *] by Def1;
A5: Rea r = r0 by A3,QUATERNI:23;
A6: Im1 r = r1 by A3,QUATERNI:23;
A7: Im2 r = r2 by A3,QUATERNI:23;
A8: q0 = jj by A1,A2,QUATERNI:12;
A9: q1 = 0 by A1,A2,QUATERNI:12;
A10: q2 = 0 by A1,A2,QUATERNI:12;
  q3 = 0 by A1,A2,QUATERNI:12;
  then r" = r0/(|.r.|^2) + (-r1/(|.r.|^2))*<i> +
  (-r2/(|.r.|^2))*<j> + (-r3/(|.r.|^2))*<k> by Th1,A4,A8,A9,A10
    .=r0/|.r.|^2 + -((r1/|.r.|^2)*<i>) +
  (-(r2/|.r.|^2))*<j> + (-(r3/|.r.|^2))*<k> by Th9
    .=r0/|.r.|^2 - r1/|.r.|^2*<i> +
  -(r2/|.r.|^2*<j>) + (-(r3/|.r.|^2))*<k> by Th9
    .=r0/|.r.|^2 - r1/|.r.|^2*<i> -
  r2/|.r.|^2*<j> + -(r3/|.r.|^2*<k>) by Th9;
  hence thesis by A3,A5,A6,A7,QUATERNI:23;
end;
