Find R* isomorphic to A, for which R is in R*.

We shall replace each element r^A Î | A | by the element r Î R ( = | R | ).

If we obtain an appropriate R* then, by isomorphism with A, there will be some b Î | R* | such that R* will satisfy S when v₁ is replaced by b.

Include in R* the following:

    i.    R* = P_R^R* : the interpretation of P_R in R*

   ii.    F* = f_F^R* : the interpretation of f_F in R*

  iii.    r* = r^R* = r

Notes

A.    1.    R may be considered as a 1-place relation on R.

               "xP_Rx is true in R, so it will be true in R*, so take R* as | R* |

        2.    R Ì R* so R is the restriction of R* to R.

        3.    R Ì R* so F is the restriction of F* to R.

B.    1.    In what follows we apply - often implicitly - a method of showing that R* or F*

              has some property that involves claiming:

              1.    R/F has that property;

              2.    The property can be expressed in L;

              3.    |=_R f Û |=_R* f  

                     Ex. 1.    The property <* on R*.    

                     Ex. 2.     The property of least upper boundedness.