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

<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#V2">admissible</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="E1:53"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&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/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">l1</font> =  <a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="termord.html#K6">Red</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>,<font color="Maroon">T</font>;<br/>
<i><font color="Green">E41</font></i>:  <a href="relset_1.html#K4">dom</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</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>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E4:53"/><i><font color="Green">E45</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">(<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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <span class="p3">(<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>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E46</font></i>: 
 <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="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E58</font></i>: 
 <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="hidden.html#R1">=</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="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E70</font></i>: 
 <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E71</font></i>: 
 <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E9:53"/><i><font color="Green">E73</font></i>: 
 <a href="card_1.html#K4">card</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, , </i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E76</font></i>: 
 <a href="card_1.html#K4">card</a> <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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T6" target="_self">Th6</a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E79</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> <span class="p3">(<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>,<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="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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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> <span class="p3">(<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>,<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="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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E3:53_2"/><i><font color="Green">E80</font></i>: 
( <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> <span class="p3">(<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>,<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="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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:53_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K3">HT</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>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:53_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="termord.html#K3">HT</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>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E6:53_2"/><i><font color="Green">E82</font></i>: 
 <a href="termord.html#K3">HT</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_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="E1:53_2_1"/><i><font color="Green">E83</font></i>: 
<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"><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>
 ;<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="E3:53_2_1"/>
<font color="Maroon">u</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</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>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E4:53_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E6:53"><i><font color="Green">E58</font></i></a>, <a class="ref" href="termord.html#T8">TERMORD:8</a></i>;<br/><b>hence </b><a NAME="E5:53_2_1"/>
<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"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="groeb_3.html#T6" target="_self">Th6</a>, , , <a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, </i>;<br/></div><b>end;</b></div><a NAME="E8:53_2"/><b>then </b>
 <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="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:53_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_3.html#T8" target="_self">Th8</a>, , <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E10:53_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:53_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E12:53_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:53_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E14:53_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E12:53"/><b>then </b><i><font color="Green">E84</font></i>: 
 <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</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> <span class="p3">(<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>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E13:53"/><i><font color="Green">E85</font></i>: 
 <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <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="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, <a class="ref" href="groeb_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E14:53"/><b>then </b>
 <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="hidden.html#R1">=</a> <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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</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> <span class="p3">(<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>,<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#T45">XBOOLE_1:45</a></i>;<br/>
<b>then </b><i><font color="Green">E86</font></i>:  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</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>,<font color="Maroon">T</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p4">(<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>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></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> <span class="p3">(<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>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="termord.html#T36">TERMORD:36</a></i>
<br/>.= 
<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>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p3">(<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>,<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#T40">XBOOLE_1:40</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>now </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="E1:53_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_4_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or  not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1"><b>case </b></a><a NAME="E1:53_4_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_4_1_1_1"/><b>then </b>
not <font color="Maroon">b</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> <span class="p3">(<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>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:53_4_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3">HT</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>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, <a class="ref" href="groeb_3.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<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="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, , <a class="txt" href="groeb_3.html#E5:53"><i><font color="Green">E46</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K6">Red</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>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, , , <a class="ref" href="termord.html#T40">TERMORD:40</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_2"><b>case </b></a><a NAME="E1:53_4_1_1_2"/><i><font color="Green">E117</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K6">Red</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>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, <a class="txt" href="groeb_3.html#E6:53"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:53_4"/>
<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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</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>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:53"/>
 <a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="funct_2.html#R2">=</a>  <a href="termord.html#K6">Red</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>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
