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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Maroon">T</font> );<br/>
<a NAME="E1:40_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">P</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:40_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">P</font></span><a href="mcart_1.html#K12">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">P</font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font>, <font color="Maroon">p2</font> = <font color="Maroon">p2</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
 <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">T</font>
;<br/>

<b>take </b>
<font color="Maroon">p1</font>
;<br/>

<b>take </b>
<font color="Maroon">p2</font>
;<br/>

<b>thus </b><a NAME="E5:40_1_1_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:40_1_1_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>

<b>thus </b><a NAME="E7:40_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:40_1_1"/><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">P</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">P</font></span><a href="mcart_1.html#K12">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p2</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:40_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="groeb_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:40_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Olive">q</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:40_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="groeb_2.html#K4" target="_self">S-Poly</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="groeb_2.html#K4" target="_self">S-Poly</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/><a NAME="E2:40_1_1_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p1</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Olive">p2</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">p1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p2</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:40_1_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E5:40_1_1_3"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">p1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p2'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:40_1_1_3"/><i><font color="Green">E43</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> &amp; <font color="Maroon">p1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E8:40_1_1_3"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><b>hence </b><a NAME="E9:40_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:40_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="groeb_2.html#K4" target="_self">S-Poly</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E8:40_1_1"/>
 <a href="groeb_2.html#K4" target="_self">S-Poly</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>


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