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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">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>







<b>set </b><font color="Maroon">p2</font> =  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>
<b>thus </b> <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">p</font>,<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> = 
<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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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">p</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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p5">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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="termord.html#T17">TERMORD:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4">|</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K28">*'</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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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="polynom7.html#T27">POLYNOM7:27</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> <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">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p0">(<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>)</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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="polynom7.html#T19">POLYNOM7:19</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="polyred.html#T4">POLYRED:4</a></i>
;<br/>

<b>thus </b> <a href="groeb_2.html#K3" target="_self">S-Poly</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">p</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">p</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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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> <span class="p5">(<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>)</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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom7.html#K4">|</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K28">*'</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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T27">POLYNOM7:27</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> <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> <span class="p0">(<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="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T19">POLYNOM7:19</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="polyred.html#T4">POLYRED:4</a></i>
;<br/>


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