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[]$[] \OT1/cmr/m/n/10 = [] = [] \OMS/cmsy/m/n/10 [ [] \OT1/cmr/m/n/10 = [] \OMS
/cmsy/m/n/10 [ []$ 
 []


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[]\OT1/ppl/m/n/10 According to Lemma[][][][][][] (in the sec-ond steps), be-cau
se of  $[] \OMS/cmsy/m/n/10 ^^R []$
 []

[7]
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[]\OT1/ppl/m/n/10 Thus, we have  $[]\OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 = []\O
ML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 = [] =
 []

[8]
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[]\OT1/ppl/m/n/10 Thus, we have  $[]\OML/cmm/m/it/10 ^^R \OT1/cmr/m/n/10 = []\O
ML/cmm/m/it/10 ^^R \OT1/cmr/m/n/10 = [][] =
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\OT1/ppl/m/n/10 Let $\OMS/cmsy/m/n/10 S$ \OT1/ppl/m/n/10 range over pre-structu
res.  Let $\OML/cmm/m/it/10 ^^P$ \OT1/ppl/m/n/10 range over $\OMS/cmsy/m/n/10 S
$\OT1/ppl/m/n/10 -semantical val-u-a-tions.  Let $[]$.
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[] \OT1/cmr/m/n/10 = [][] [] []\OML/cmm/m/it/10 :
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\OT1/ppl/m/n/10 where the one but last equa-tion holds by the \OT1/ppl/m/sc/10 
Ex-plic-it-ness Lemma \OT1/ppl/m/n/10 be-cause of  []
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[]$ 
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[]\OT1/ppl/m/n/10 For first-order com-plete-ness we may need dif-fer-ent terms 
$\OML/cmm/m/it/10 t$ \OT1/ppl/m/n/10 clos-ing dif-fer-ent branches
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\OT1/ppl/m/n/10 The en-try in the choice-condition will re-strict fu-ture sub-s
ti-tu-tions of $\OML/cmm/m/it/10 X$ \OT1/ppl/m/n/10 to sat-isfy Hilbert's
 []


Overfull \hbox (4.7829pt too wide) in paragraph at lines 3332--3347
[]\OT1/ppl/m/n/10 The fol-low-ing rules come with pro-viso that  $\OML/cmm/m/it
/10 X$ \OT1/ppl/m/n/10 and $\OML/cmm/m/it/10 a$ \OT1/ppl/m/n/10 have the same t
ype;  []
 []

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Overfull \hbox (6.43602pt too wide) in paragraph at lines 3870--3909
\OT1/ppl/m/n/10 have  $[] \OT1/cmr/m/n/10 = []$\OT1/ppl/m/n/10 ,  and thus  $[]
 \OT1/cmr/m/n/10 =
 []


Overfull \hbox (9.75047pt too wide) in paragraph at lines 3910--3943
\OT1/ppl/m/n/10 ously the case if  $[] \OT1/cmr/m/n/10 = []$\OT1/ppl/m/n/10 .  
Oth-er-wise, we have  $[] \OT1/cmr/m/n/10 =
 []


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\OT1/ppl/m/sc/10 tion \OT1/ppl/m/n/10 [\OT1/ppl/m/sc/10 Value\OT1/ppl/m/n/10 ] 
\OT1/ppl/m/sc/10 Lemma\OT1/ppl/m/n/10 , it suf-fices to show  $[] \OT1/cmr/m/n/
10 =
 []


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[]$\OT1/ppl/m/n/10 ,  but this is the case by the for-ward di-rec-tion of the $
\OMS/cmsy/m/n/10 8$\OT1/ppl/m/sc/10 -Lemma \OT1/ppl/m/n/10 be-cause, for [],
 []

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Package natbib Warning: There were undefined citations.

(./proof-search.aux)

LaTeX Warning: There were undefined references.

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Here is how much of TeX's memory you used:
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Output written on proof-search.dvi (29 pages, 210092 bytes).