<?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  non <a href="struct_0.html#V3">empty</a> <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="vectsp_1.html#V10">degenerated</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">p</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>;<br/>







<b>assume </b><a NAME="E1:3"/>
<font color="Maroon">f</font> <a href="polyred.html#R4">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">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R3">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><a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E42</font></i>: 
<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="polyred.html#D5">POLYRED:def 5</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>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="E7:3"/><i><font color="Green">E43</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="pboole.html#R6">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</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">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E8:3"/><i><font color="Green">E44</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">*'</a> <font color="Maroon">p</font></span>)</span> <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="polyred.html#T22">POLYRED:22</a></i>;<br/>
<a NAME="E9:3"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="polyred.html#T39">POLYRED:39</a></i>;<br/>
<b>set </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>;<br/>
<a NAME="E10:3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E11:3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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/>
<a NAME="E12:3"/>
 <a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/>
<a NAME="E13:3"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/>
<a NAME="E14:3"/>
<span class="p1">(<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="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><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> <a href="vectsp_1.html#K4">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>;<br/>
<a NAME="E15:3"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<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="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E16:3"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<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="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> is <a href="rlvect_1.html#V8">non-zero</a>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<a NAME="E17:3"/>
 <a href="polynom7.html#K3">coefficient</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>;<br/>
<a NAME="E18:3"/><b>then </b>
 <a href="termord.html#K4">HC</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>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>;<br/>
<a NAME="E19:3"/><b>then </b>
 <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> <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="termord.html#T17">TERMORD:17</a></i>;<br/>
<b>then 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="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</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/>
<i><font color="Green">E50</font></i>:  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="termord.html#T31">TERMORD:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
<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>
<b>by </b><i>, <a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
;<br/>
<a NAME="E22:3"/><b>then </b>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E23:3"/><b>then </b>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<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/>
<b>hence </b><a NAME="E24:3"/>
 ex <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span> &amp; not  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, , , </i>;<br/>


</div>
