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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#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#V3">Abelian</a> <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> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">f1</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <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>assume </b><a NAME="E1:62"/><i><font color="Green">E39</font></i>: 
(  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>,<font color="Maroon">f1</font> &amp; ( for <font color="Olive">b1</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</font> holds <br/> not  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b1</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<font color="Olive">q</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>;<br/>
<a NAME="E4:62"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i>, <a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 ;<br/><b>assume </b><a NAME="E2:62_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><a NAME="E3:62_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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="groeb_3.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:62_1"/><i><font color="Green">E70</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<font color="Maroon">h</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:62_1"><i><font color="Green">E58</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:62_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:62_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:62_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:62_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E6:62_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font>,<span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><a NAME="E7:62_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:62_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><a NAME="E9:62_1_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<font color="Maroon">q</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/><b>hence </b><a NAME="E10:62_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<font color="Maroon">q</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:62_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:62"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(<a class="ref" href="groeb_3.html#T8" target="_self">Th8</a>, );<br/></i>
<a NAME="E7:62"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<a NAME="E8:62"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>hence </b><a NAME="E9:62"/>
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="groeb_1.html#K1">}</a></span>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>,<font color="Maroon">f1</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>


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