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*') = (Rea z)^2+(Im1 z)^2+(Im2 z)^2+(Im3 z)^2 &
  Im1(z*z*') = 0 & Im2(z*z*') = 0 & Im3(z*z*') = 0
proof
A1: z = [*Rea z, Im1 z, Im2 z, Im3 z*] by Th17;
  z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th36;
  then z*z*' = [* Rea z*Rea z-(Im1 z)*(-Im1 z)-Im2 z*(-Im2 z)-(Im3 z)*(-Im3
  z), Rea z*(-Im1 z)+Im1 z*Rea z+Im2 z*(-Im3 z)-(Im3 z)*(-Im2 z),
  Rea z*(-Im2 z)+Rea z*Im2 z+(-Im1 z)*(Im3 z)-(-Im3 z)*Im1 z,
  Rea z*(-Im3 z)+(Im3 z)*Rea z+Im1 z*(-Im2 z)-Im2 z*(-Im1 z) *] by A1,Def9
    .= [* (Rea z)^2+(Im1 z)^2+(Im2 z)^2+(Im3 z)^2,0,0,0 *];
  hence thesis by Th16;
end;
