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

theorem
  <k>"=-<k>
proof
A1: Im3 <k>"= - (Im3 <k>) / (|.<k>.|^2) by QUATERN2:29
    .= -1 by QUATERNI:31;
A2: Im2 <k>" = -(Im2 <k>) / (|.<k>.|^2) by QUATERN2:29
    .= 0 by QUATERNI:31;
A3: Im1 <k>" = -(Im1 <k>) / (|.<k>.|^2) by QUATERN2:29
    .= 0 by QUATERNI:31;
 Rea <k>" = (Rea <k>) / (|.<k>.|^2) by QUATERN2:29
    .= 0 by QUATERNI:31;
then <k>"=[*-0,-0,-0,-jj*] by A3,A2,A1,QUATERNI:24
    .=-<k> by QUATERN2:4;
  hence thesis;
end;
