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

theorem
  Rea r" = (Rea r) / (|.r.|^2) & Im1 r" = - (Im1 r) / (|.r.|^2) &
  Im2 r" = - (Im2 r) / (|.r.|^2) &
  Im3 r" = - (Im3 r) / (|.r.|^2)
proof
  consider q0,q1,q2,q3,r0,r1,r2,r3 being Element of REAL such that
A1: 1q = [*q0,q1,q2,q3*] and
A2: r = [*r0,r1,r2,r3*] and
A3: 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;
A4: 1q=[*jj,In(0,REAL)*] by ARYTM_0:def 5
    .=[*jj,0,0,0*] by QUATERNI:91; then
A5: q0 = jj by A1,QUATERNI:12;
A6: q1 = 0 by A1,A4,QUATERNI:12;
A7: q2 = 0 by A1,A4,QUATERNI:12;
A8: q3 = 0 by A1,A4,QUATERNI:12;
A9: Rea r = r0 by A2,QUATERNI:23;
A10: Im1 r = r1 by A2,QUATERNI:23;
A11: Im2 r = r2 by A2,QUATERNI:23;
A12: Im3 r = r3 by A2,QUATERNI:23;
  r" = [*r0/(|.r.|^2), -r1/(|.r.|^2),
  -r2/(|.r.|^2), -r3/(|.r.|^2) *] by A3,A5,A6,A7,A8;
  hence thesis by A9,A10,A11,A12,QUATERNI:23;
end;
