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

theorem
  z is Real implies z*<k>=[*0,0,0,Rea z*]
proof
  assume
A1: z is Real;
  then reconsider x = z as Real;
A2: x = Rea z by Lm1;
  z*<k>=[*Rea (z*<k>),Im1 (z*<k>),Im2 (z*<k>),Im3 (z*<k>)*] by QUATERNI:24
    .=[*0,Im1 (z*<k>),Im2 (z*<k>),Im3 (z*<k>)*] by A1,QUATERNI:35
    .=[*0,0,Im2 (z*<k>),Im3 (z*<k>)*] by A1,QUATERNI:35
    .=[*0,0,0,Im3 (z*<k>)*] by A1,QUATERNI:35
    .=[*0,0,0,Rea z*] by A2,QUATERNI:35;
  hence thesis;
end;
