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
  z = Rea z+(Im1 z)*<i> + (Im2 z)*<j> + (Im3 z)*<k>
proof
  [*Rea z, Im1 z, Im2 z, Im3 z*] = z by Th17;
  hence thesis by Lm19;
end;
