reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

theorem
  Im1 z = 0 & Im2 z = 0 & Im3 z = 0 implies z*' = z
proof
  assume that
A1: Im1 z = 0 and
A2: Im2 z = 0 and
A3: Im3 z = 0;
  reconsider Rz = Rea z, zz = 0 as Element of REAL by XREAL_0:def 1;
A4: z = [*Rea z,0,0,0*] by A1,A2,A3,Th17
    .= [*Rz,zz*] by Lm3
    .= Rea z by ARYTM_0:def 5;
  z*' = [*Rea z, 0,0,0*] by A1,A2,A3,Th36
    .= [*Rz, zz*] by Lm3
    .= Rea z by ARYTM_0:def 5;
  hence thesis by A4;
end;
