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**Example text**

CN ; b1 , . . , bn ) < 0, contradiction. D. 2 Ultraproducts The notion of an ultraproduct of certain mathematical structures is a very useful tool introduced in model theory. We shall deal here only with ultraproducts of ordered fields. 11). 6 in the next section, and obtain effectivity results in Chapter 8. Let S = ∅ be an arbitrary index set. 6), footnote 4: K (n) denotes the n-fold Cartesian product K × K × · · · × K; and K m denotes { xm | x ∈ K }. The superscript [s] in K [s] is used here only as an index, and should not be confused with the earlier superscripts.

An ∈ φ[s] K [s] ∈ F. 5). 1). Observing that ∀y ψ may be replaced by the equivalent ¬ ∃y ¬ψ, the following induction step remains to be shown. Let φ(X1 , . . , Xn ) be given as ∃y ψ(X1 , . . , Xn , y) with ψ prenex, and suppose (using induction on the length of φ) that for all b := b[s] ∈ K [s] we have already shown K [s] F ⇐⇒ ([a1 ], . . , [an ], [b]) ∈ ψ [s] [s] [s] K [s] s (a1 , . . , a[s] n ,b ) ∈ ψ ∈ F; we show the equivalence for φ. (⇒): From [s] s [s] a1 , . . , a[s] ∈ ψ [s] (X1 , .

W A S S 0 cd wT cv T Aw ST AS 2 cd + wT Aw cdv T + wT AS = cdv + S T Aw cvv T + S T AS . Comparing blocks, cd2 + wT Aw = c =⇒ wT Aw = c − cd2 = c(1 − d2 ), cvv T + S T AS = B =⇒ S T AS = B − cvv T , wT AS = −cdv T , and S T Aw = −cdv. Let Q = S +λwv T , where λ ∈ K will be chosen later. Then (using the above), QT AQ = (S T + λvwT )A(S + λwv T ) = S T AS + λS T Awv T + λvwT AS + λ2 vwT Awv T = B − cvv T − λcdvv T − λcdvv T + λ2 c(1 − d2 )vv T = B + (λ2 (1 − d2 ) − 2dλ − 1)cvv T .