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

theorem
  Im1 z = 0 & Im2 z = 0 & Im3 z = 0 implies z = Rea z
proof
  set x = Rea z;
  assume Im1 z=0 & Im2 z = 0 & Im3 z = 0;
  then A1: z=[*x,0,0,0*] by QUATERNI:24;
  reconsider xx=x, zz=0 as Element of REAL by XREAL_0:def 1;
  [*x,0,0,0*]=[*xx,zz*] by QUATERNI:91;
  hence thesis by A1,ARYTM_0:def 5;
end;
