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
  z1 - z2 = Rea z1 - Rea z2 + (Im1 z1 - Im1 z2)*<i> +
    (Im2 z1 - Im2 z2)*<j> + (Im3 z1 - Im3 z2)*<k> by Lm25;
