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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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">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>assume </b><a NAME="E1:28"/><i><font color="Green">E40</font></i>: 
 <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="groeb_1.html#K3" target="_self">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>;<br/>
<a NAME="E2:28"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">f</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Olive">f</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> holds <br/><font color="Olive">f</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<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>;<br/>

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">f</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> )
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Olive">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> ) </span>}</span> 
 
;<br/>
<a NAME="E3:28_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font>
 
;<br/>
<a NAME="E4:28_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K3" target="_self">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:28_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> :  ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:28_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> &amp;  ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> ) )
 ;<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="E9:28_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">b'</font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:28_1"/>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
<b>by </b><i/>;<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="E12:28_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">b'</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <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> )
 ;<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="E14:28_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></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>;<br/>
<a NAME="E15:28_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/>
<a NAME="E16:28_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<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="termord.html#D6">TERMORD:def 6</a></i>;<br/>
<a NAME="E17:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R3">reduces_to</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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></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>,<font color="Maroon">p</font>, <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="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E18:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R4">reduces_to</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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></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>,<font color="Maroon">p</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="E19:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R5">reduces_to</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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></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>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/>
<b>hence </b><a NAME="E20:28_1"/>
<font color="Maroon">f</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">f</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<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>;<br/>

<b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> or <font color="Maroon">f</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>, <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="rewrite1.html#T13">REWRITE1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</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>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:28_2_1_2"/><i><font color="Green">E73</font></i>: 
not  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><a NAME="E3:28_2_1_2"/>
<font color="Maroon">f</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">v</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="E5:28_2_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">v</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><a NAME="E6:28_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><a NAME="E7:28_2_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E8:28_2_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rewrite1.html#R9">has_a_normal_form_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D14">REWRITE1:def 14</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:28_2_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">g</font> <a href="rewrite1.html#R4">is_a_normal_form_of</a> <font color="Maroon">f</font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D11">REWRITE1:def 11</a></i>;<br/><a NAME="E11:28_2_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">g</font> <a href="rewrite1.html#R3">is_a_normal_form_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/><a NAME="E12:28_2_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">g</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="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font>, <font color="Maroon">gg</font> = <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">ff</font> = <font color="Maroon">ff</font>, <font color="Maroon">gg</font> = <font color="Maroon">gg</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/><a NAME="E16:28_2_1_2"/>
<font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">gg</font>
 <b>by </b><i/>;<br/><a NAME="E17:28_2_1_2"/><b>then </b>
<font color="Maroon">ff</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">gg</font> <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="polyred.html#T59">POLYRED:59</a></i>;<br/><a NAME="E18:28_2_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">gg</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ff</font> <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#T16">IDEAL_1:16</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">gg</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ff</font> = 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ff</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">ff</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">ff</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</font>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
;<br/><a NAME="E20:28_2_1_2"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">gg</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#T14">IDEAL_1:14</a></i>;<br/><a NAME="E21:28_2_1_2"/><b>then </b>
<font color="Maroon">g</font> <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="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/><a NAME="E22:28_2_1_2"/><b>then </b>
<font color="Maroon">g'</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">u</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="E24:28_2_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">g'</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">u</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><a NAME="E25:28_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/><b>hence </b><a NAME="E26:28_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D5">REWRITE1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_3"/><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 ;<br/><b>then consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">a'</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">b'</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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><i><font color="Green">E82</font></i>: <font color="Maroon">b'</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>consider </b><font color="Maroon">aa</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:28_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">aa</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c'</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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">aa</font>,<font color="Maroon">c'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><i><font color="Green">E95</font></i>: <font color="Maroon">c'</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font>, <font color="Maroon">c'</font> = <font color="Maroon">c'</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="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b'</font>, <font color="Maroon">cc</font> = <font color="Maroon">c'</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">bb</font> = <font color="Maroon">bb</font>, <font color="Maroon">cc</font> = <font color="Maroon">cc</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/><a NAME="E11:28_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">b'</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">cc</font>
 <b>by </b><i/>;<br/><a NAME="E12:28_3"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/><a NAME="E13:28_3"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T32">REWRITE1:32</a></i>;<br/><a NAME="E14:28_3"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/><a NAME="E15:28_3"/><b>then </b>
<font color="Maroon">bb</font>,<font color="Maroon">cc</font> <a href="polyred.html#R11">are_congruent_mod</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, , <a class="ref" href="polyred.html#T57">POLYRED:57</a></i>;<br/><a NAME="E16:28_3"/><b>then </b>
<font color="Maroon">bb</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">cc</font> <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="polyred.html#D14">POLYRED:def 14</a></i>;<br/><a NAME="E17:28_3"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">b'</font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E18:28_3"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R5">are_convergent_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="polyred.html#T50">POLYRED:50</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:28"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="rewrite1.html#D24">REWRITE1:def 24</a></i>;<br/>


</div>
