<?xml version="1.0"?>
<div class="add">

<a NAME="E1:27_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="polyred.html#E1:27_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span> ;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b'</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 ;<br/><b>thus </b><a NAME="E4:27_1_1_1"/>
 ex <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="polyred.html#E3:27_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_2"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span>  )
 ;<br/><b>then consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:27_1_1_2"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 ;<br/><b>consider </b><font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:27_1_1_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b2</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><font color="Maroon">b1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="polyred.html#E1:27_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b2</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><b>hence </b><font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="polyred.html#E1:27_1_1_2"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_3"/>
<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>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_3"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">b'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:27_1_1_3"/><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b'</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><font color="Maroon">u</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b'</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E7:27_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E3:27_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u'</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">u'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b'</font>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:27_1_1_4"/>
<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>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:27_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E8:27_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E9:27_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_5"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E2:27_1_1_5"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="polyred.html#E2:27_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">b'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:27_1_1_5"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 ;<br/><a NAME="E5:27_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:27_1_1_5"/>
<font color="Maroon">u</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="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:27_1_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:27_1_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:27_1_1"/>
<font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font> is <a href="polynom1.html#V2">finite-Support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>


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