<?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#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#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</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">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be  <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>;<br/><b>let </b><font color="Maroon">g</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">f'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>, <font color="Maroon">g'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/>













<b>assume </b><a NAME="E1:93"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> )
 ;<br/>

<b>assume </b><a NAME="E2:93"/>
<font color="Maroon">f</font> <a href="polyred.html#R5" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</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="E4:93"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f</font> <a href="polyred.html#R4" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:93"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R3" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E9:93"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E6"><i><font color="Green">Lemma72</font></i></a>, </i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g</font> = 
<font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><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="E1:93_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:93_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></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"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E3:93_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:93_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:93"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:93_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:93_3"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:93_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:93_3"/><b>then </b>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:93_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="group_1.html#K10">*</a> <font color="Maroon">p'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span>)</span> = 
<font color="Maroon">m</font> <a href="group_1.html#K10">*</a> <font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>hence </b><a NAME="E11:93_3"/>
 ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font> st <font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">v</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a></i>;<br/>
<a NAME="E16:93"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E17:93"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="binom.html#T3">BINOM:3</a></i>;<br/>
<a NAME="E18:93"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E19:93"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">g</font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<a NAME="E23:93"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T19">RLVECT_1:19</a></i>;<br/>
<font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g</font> = 
<font color="Maroon">f</font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">g'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E25:93"/>
<font color="Maroon">f'</font>,<font color="Maroon">g'</font> <a href="polyred.html#R11" target="_self">are_congruent_mod</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, , </i>;<br/>


</div>
