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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> non <a href="realset2.html#V3">trivial</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:35"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:35_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:35_1"/>
contradiction
 <b>by </b><i><a class="ref" href="groeb_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:35"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="polynom7.html#V1">non-zero</a> <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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">l</font> =  <a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">sl</font> =  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>;<br/>
<a NAME="E5:35"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E6:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E7:35"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="termord.html#T36">TERMORD:36</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E45</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 ;<br/><a NAME="E2:35_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E3:35_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:35_2"/>
contradiction
 <b>by </b><i><a class="ref" href="groeb_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_3"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span>;<br/><a NAME="E3:35_3"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:35_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:35"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> = 
 <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
<br/>.= 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="groeb_3.html#T1" target="_self">Th1</a>, </i>
;<br/>
<b>hence </b><a NAME="E12:35"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="groeb_3.html#T1" target="_self">Th1</a>, </i>;<br/>


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