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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">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">G</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="E1:38"/><i><font color="Green">E31</font></i>: 
not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">g1</font>,<font color="Olive">g2</font>,<font color="Maroon">T</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g1</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g2</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="E1:38_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 ;<br/><b>thus </b><a NAME="E2:38_1"/>
for <font color="Olive">m1</font>, <font color="Olive">m2</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st  <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Olive">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Olive">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>,<font color="Maroon">T</font> holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Olive">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Olive">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <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="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">m2</font> be   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>,<font color="Maroon">T</font>
 ;<br/>

<b>set </b><font color="Maroon">t1</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">t2</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">a1</font> =  <a href="termord.html#K4">HC</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">a2</font> =  <a href="termord.html#K4">HC</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">b1</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font>;<br/>
<b>set </b><font color="Maroon">b2</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font>;<br/>
<b>set </b><font color="Maroon">u1</font> =  <a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font>;<br/>
<b>set </b><font color="Maroon">u2</font> =  <a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font>;<br/>
<a NAME="E2:38_1_1"/><i><font color="Green">E37</font></i>: 
(  <a href="termord.html#K4">HC</a> <font color="Maroon">g1</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> &amp;  <a href="termord.html#K4">HC</a> <font color="Maroon">g2</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/>
<b>then reconsider </b><font color="Maroon">a1</font> =  <a href="termord.html#K4">HC</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font>, <font color="Maroon">a2</font> =  <a href="termord.html#K4">HC</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font> as  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>, <font color="Maroon">g2</font> = <font color="Maroon">g2</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/>
<i><font color="Green">E39</font></i>:  <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">T</font> = 
 <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T22">TERMORD:22</a></i>
<br/>.= 
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="termord.html#T22">TERMORD:22</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_1"/>
(  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or (  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp;  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_1"><b>case </b></a><a NAME="E1:38_1_1_2_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_1_1_1_1"/>
(  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:38_1_1_2_1_1_1_1_1"/>
 <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1_2_1_1_1_1_1"/><b>then </b>
 <a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</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="E3:38_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</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/><a NAME="E4:38_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</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#T5">POLYRED:5</a></i>;<br/><a NAME="E5:38_1_1_2_1_1_1_1_1"/><b>then </b>
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</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="E6:38_1_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</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/><a NAME="E7:38_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</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#T4">POLYRED:4</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_2_1_1_1_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <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="38_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:38_1_1_2_1_1_1_1_2"/>
 <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1_2_1_1_1_1_2"/><b>then </b>
 <a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</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="E3:38_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m2</font> <a href="hidden.html#R1">=</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/><a NAME="E4:38_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</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#T5">POLYRED:5</a></i>;<br/><a NAME="E5:38_1_1_2_1_1_1_1_2"/><b>then </b>
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</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="E6:38_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</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/><a NAME="E7:38_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R1">=</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#T4">POLYRED:4</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_2_1_1_1_1_2"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_2_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_2"><b>case </b></a><a NAME="E1:38_1_1_2_1_2"/><i><font color="Green">E43</font></i>: 
(  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp;  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b1</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font>, <font color="Maroon">b2</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font> as  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:38_1_1_2_1_2"/>
(  <a href="termord.html#K4">HC</a> <font color="Maroon">m1</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> &amp;  <a href="termord.html#K4">HC</a> <font color="Maroon">m2</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#T23">TERMORD:23</a></i>;<br/><a NAME="E4:38_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">m1</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> &amp; <font color="Maroon">m2</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">m1</font> = <font color="Maroon">m1</font>, <font color="Maroon">m2</font> = <font color="Maroon">m2</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">E47</font></i>:  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b1</font>,<span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a1</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T3">TERMORD:3</a></i>
<br/>.= 
<font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a1</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a1</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D8">TERMORD:def 8</a></i>
<br/>.= 
 <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="termord.html#T33">TERMORD:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T33">TERMORD:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a2</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D8">TERMORD:def 8</a></i>
<br/>.= 
<font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a2</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b2</font>,<span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a2</font>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T3">TERMORD:3</a></i>
;<br/><a NAME="E7:38_1_1_2_1_2"/>
( <font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</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="groeb_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><a NAME="E8:38_1_1_2_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font> is <a href="rlvect_1.html#V8">non-zero</a> &amp; <font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font> 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/><b>then </b><i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span> = 
 <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</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">g2</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="E10:38_1_1_2_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E12:38_1_1_2_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">s1</font> <a href="pboole.html#R6">=</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E13:38_1_1_2_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">s2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E15:38_1_1_2_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font> <a href="pboole.html#R6">=</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E16:38_1_1_2_1_2"/>
(  <a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E17:38_1_1_2_1_2"/><b>then </b>
 <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#R3">divides</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E19:38_1_1_2_1_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E20:38_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</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">g1</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s1</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">g1</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="groeb_2.html#E1:38_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom1.html#T39">POLYNOM1:39</a></i>;<br/><b>then </b><i><font color="Green">E54</font></i>:  <a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s1</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
;<br/><a NAME="E22:38_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</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">g2</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</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">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="groeb_2.html#E2:38"><i><font color="Green">E33</font></i></a>, <a class="txt" href="groeb_2.html#E1:38_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom1.html#T39">POLYNOM1:39</a></i>;<br/><b>then </b><i><font color="Green">E55</font></i>:  <a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
;<br/><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font> = 
 <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"><font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <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/>.= 
<font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/><b>then </b><i><font color="Green">E56</font></i>: <font color="Maroon">b2</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</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"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
;<br/><a NAME="E27:38_1_1_2_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E28:38_1_1_2_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">s1</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E29:38_1_1_2_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a>  <a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E30:38_1_1_2_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">s2</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E2:38"><i><font color="Green">E33</font></i></a>, </i>;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b1</font>,<span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b1</font>,<span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b2</font>,<span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s1</font></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b2</font>,<span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:38_1_1"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">b2</font>,<span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T18">POLYRED:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s2</font></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T18">POLYRED:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <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">b2</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T9">POLYRED:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></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">b2</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></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">b2</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E2:38_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="polyred.html#T11">POLYRED:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T11">POLYRED:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</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">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p4">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</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">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T9">POLYRED:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</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">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="polynom7.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="polynom7.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">g2</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">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="polyred.html#K1">*'</a> <span class="p2">(<span class="default"><a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a>, , <a class="txt" href="groeb_2.html#E5:38_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="vectsp_1.html#K5">/</a> <font color="Maroon">a2</font></span>)</span>,<font color="Maroon">v</font></span>)</span> <a href="polynom1.html#K29">*'</a> <span class="p1">(<span class="default"><a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
;<br/><a NAME="E34:38_1_1_2_1_2"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</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="E35:38_1_1_2_1_2"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <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#T48">POLYRED:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38_1_1"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">g2</font></span>)</span>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon">G</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


</div>
