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

theorem
  z is Real implies z*<j>=[*0,0,Rea z,0*]
proof
  assume
A1: z is Real;
  then reconsider x = z as Real;
A2: x = Rea z by Lm1;
  z*<j>=[*Rea (z*<j>),Im1 (z*<j>),Im2 (z*<j>),Im3 (z*<j>)*] by QUATERNI:24
    .=[*0,Im1 (z*<j>),Im2 (z*<j>),Im3 (z*<j>)*] by A1,QUATERNI:34
    .=[*0,0,Im2 (z*<j>),Im3 (z*<j>)*] by A1,QUATERNI:34
    .=[*0,0,x,Im3 (z*<j>)*] by QUATERNI:34
    .=[*0,0,Rea z,0*] by A2,QUATERNI:34;
  hence thesis;
end;
