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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">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/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <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> or <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> or ( <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> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green">E31</font></i>: 
<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>
 ;<br/><div class="add"><i><font color="Green">E33</font></i>:  <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">T</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"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
<br/>.= 
 <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>thus </b> <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">m1</font>,<font color="Maroon">m2</font>,<font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <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>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m2</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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T87">POLYNOM1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p4">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m2</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<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> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="termord.html#T17">TERMORD:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T5">POLYRED:5</a></i>
<br/>.= 
 <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="polynom1.html#T83">POLYNOM1:83</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green">E34</font></i>: 
<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>
 ;<br/><div class="add"><i><font color="Green">E35</font></i>:  <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">T</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"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
<br/>.= 
 <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>thus </b> <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">m1</font>,<font color="Maroon">m2</font>,<font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m1</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> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <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>)</span>
<b>by </b><i>, <a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T87">POLYNOM1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p4">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<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> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m1</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="termord.html#T17">TERMORD:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T5">POLYRED:5</a></i>
<br/>.= 
 <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="polynom1.html#T83">POLYNOM1:83</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_3"><b>suppose </b></a><a NAME="E1:32_1_3"/>
( <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> )
 ;<br/><div class="add"><a NAME="E2:32_1_3"/><b>then </b>
(  <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#T17">TERMORD:17</a></i>;<br/><a NAME="E3:32_1_3"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font> is <a href="rlvect_1.html#V8">non-zero</a> &amp;  <a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</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/><i><font color="Green">E39</font></i>: <font color="Maroon">m1</font> = 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></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"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m1</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
;<br/><i><font color="Green">E40</font></i>: <font color="Maroon">m2</font> = 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></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"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m2</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
<br/>.= 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T23">TERMORD:23</a></i>
;<br/><a NAME="E6:32_1_3"/><i><font color="Green">E41</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">m1</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">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</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 href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<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">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E8:32_1_3"/><i><font color="Green">E43</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">m2</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">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">m2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">m2</font>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</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 href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m1</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<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">m1</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m2</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>thus </b><a NAME="E10:32_1_3"/>
 <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">m1</font>,<font color="Maroon">m2</font>,<font color="Maroon">T</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="groeb_2.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom1.html#T83">POLYNOM1:83</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
