reserve z1,z2,z3,z4,z for Quaternion;

theorem
  (z*')" = (z")*'
proof
 Im1 (z*')" = - (Im1 z*') / (|.z*'.|^2) by QUATERN2:29;
then A1: Im1 (z*')" = - (-Im1 z) / (|.z*'.|^2) by QUATERNI:44;
 Im2 (z*')" = - (Im2 z*') / (|.z*'.|^2) by QUATERN2:29;
then A2: Im2 (z*')" = - (-Im2 z) / (|.z*'.|^2) by QUATERNI:44;
 Im2 (z")*' = -Im2 z" by QUATERNI:44;
then Im2 (z")*' = -(-(Im2 z) / (|.z.|^2)) by QUATERN2:29;
then A3: Im2 (z")*' = -(-(Im2 z) / (|.z*'.|^2)) by QUATERNI:73;
 Im1 (z")*' = -Im1 z" by QUATERNI:44;
then Im1 (z")*' = -(-(Im1 z) / (|.z.|^2)) by QUATERN2:29;
then A4: Im1 (z")*' = -(-(Im1 z) / (|.z*'.|^2)) by QUATERNI:73;
 Im3 (z")*' = -Im3 z" by QUATERNI:44;
then Im3 (z")*' = -(-(Im3 z) / (|.z.|^2)) by QUATERN2:29;
then A5: Im3 (z")*' = -(-(Im3 z) / (|.z*'.|^2)) by QUATERNI:73;
 Rea (z")*' = Rea z" by QUATERNI:44;
then Rea (z")*' = (Rea z) / (|.z.|^2) by QUATERN2:29;
then A6: Rea (z")*' = (Rea z) / (|.z*'.|^2) by QUATERNI:73;
 Im3 (z*')" = - (Im3 z*') / (|.z*'.|^2) by QUATERN2:29;
then A7: Im3 (z*')" = - (-Im3 z) / (|.z*'.|^2) by QUATERNI:44;
 Rea (z*')" = (Rea z*') / (|.z*'.|^2) by QUATERN2:29;
then Rea (z*')" = (Rea z) / (|.z*'.|^2) by QUATERNI:44;
  hence thesis by A1,A2,A7,A6,A4,A3,A5,QUATERNI:25;
end;
