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
  Rea(z - z*') = 0 & Im1(z - z*') = 2*Im1 z &
  Im2(z - z*') = 2*Im2 z & Im3(z - z*') = 2*Im3 z
proof
A1: z = [*Rea z, Im1 z, Im2 z, Im3 z*] by Th17;
A2: z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th36;
A3: Im1 z*' = - Im1 z by A2,Th16;
A4: Im2 z*' = - Im2 z by A2,Th16;
A5: Im3 z*' = - Im3 z by A2,Th16;
  set zz = z*';
  -zz = [*-Rea zz, -Im1 zz, -Im2 zz, -Im3 zz*] by Th55;
  then -zz = [*-Rea z,Im1 z,Im2 z,Im3 z*] by A2,Th16,A3,A4,A5;
  then z - zz = [*Rea z+ (-Rea z), Im1 z + Im1 z, Im2 z + Im2 z, Im3 z +Im3
  z *] by A1,Def6
    .= [*0, 2* Im1 z, 2* Im2 z,2* Im3 z*];
  hence thesis by Th16;
end;
