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

theorem
  2 * Rea z = Rea (z+ z*')
proof
  z = [*Rea z, Im1 z, Im2 z, Im3 z*] & z*' = [*Rea z, -Im1 z, -Im2 z, -Im3
    z*] by QUATERNI:24,43;
  then z + z*' = [*Rea z+Rea z, Im1 z+ -Im1 z, Im2 z+ -Im2 z, Im3 z+ -Im3 z*]
  by QUATERNI:def 7
    .= [*2*Rea z,0,0,0*];
  hence thesis by QUATERNI:23;
end;
