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

theorem
  |.z1.|=|.z2.| & |.z1.| <>0 & z1"=z2" implies z1=z2
proof
  assume that
A1: |.z1.|=|.z2.| and
A2: |.z1.| <>0 and
A3: z1"=z2";
A4: |.z1.|^2 <>0 by A2,XCMPLX_1:6;
  Im3 z1" = -(Im3 z1)/(|.z1.|^2) by QUATERN2:29;
  then -(Im3 z1)/(|.z1.|^2)= -(Im3 z2)/(|.z2.|^2) by A3,QUATERN2:29;
  then A5: (Im3 z1)= (Im3 z2) by A1,A4,XCMPLX_1:53;
  Im2 z1" = -(Im2 z1)/(|.z1.|^2) by QUATERN2:29;
  then -(Im2 z1)/(|.z1.|^2)= -(Im2 z2)/(|.z2.|^2) by A3,QUATERN2:29;
  then A6: (Im2 z1)= (Im2 z2) by A1,A4,XCMPLX_1:53;
  Im1 z1" = -(Im1 z1)/(|.z1.|^2) by QUATERN2:29;
  then -(Im1 z1)/(|.z1.|^2)= -(Im1 z2)/(|.z2.|^2) by A3,QUATERN2:29;
  then A7: (Im1 z1)= (Im1 z2) by A1,A4,XCMPLX_1:53;
  Rea z1" = Rea z1/(|.z1.|^2) by QUATERN2:29;
  then Rea z1/(|.z1.|^2) = Rea z2/(|.z2.|^2) by A3,QUATERN2:29;
  then Rea z1= Rea z2 by A1,A4,XCMPLX_1:53;
  hence thesis by A7,A6,A5,QUATERNI:25;
end;
