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

theorem
  z is Real implies z*<i>=[*0,Rea z,0,0*]
proof
  assume
A1: z is Real;
  then reconsider x = z as Real;
A2: x = Rea z by Lm1;
  z*<i>=[*Rea (z*<i>),Im1 (z*<i>),Im2 (z*<i>),Im3 (z*<i>)*] by QUATERNI:24
    .=[*0,Im1 (z*<i>),Im2 (z*<i>),Im3 (z*<i>)*] by A1,QUATERNI:33
    .=[*0,x,Im2 (z*<i>),Im3 (z*<i>)*] by QUATERNI:33
    .=[*0,x,0,Im3 (z*<i>)*] by QUATERNI:33
    .=[*0,Rea z,0,0*] by A2,QUATERNI:33;
  hence thesis;
end;
