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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <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="E1:57"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/>
<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="groeb_1.html#E9:57"><i><font color="Green">E43</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/><b>hence </b><a NAME="E3:57_1"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  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>set </b><font color="Maroon">q</font> = <font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font>;<br/>
<b>take </b>
<font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font>
;<br/>

<b>reconsider </b><font color="Maroon">pp</font> = <font color="Maroon">p</font>, <font color="Maroon">cc</font> = <font color="Maroon">c</font> <a href="polynom7.html#K4">|</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pp</font> = <font color="Maroon">pp</font>, <font color="Maroon">cc</font> = <font color="Maroon">cc</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K4">|</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom7.html#T27">POLYNOM7:27</a></i>
<br/>.= 
<font color="Maroon">cc</font> <a href="group_1.html#K10">*</a> <font color="Maroon">pp</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<b>hence </b><a NAME="E8:57"/>
<font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:57"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a></i>;<br/>

<a NAME="E9:57"/><i><font color="Green">E43</font></i>: 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="polyred.html#T21">POLYRED:21</a></i>;<br/>
<b>then </b> <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="groeb_1.html#E9:57"><i><font color="Green">E43</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/>
<b>hence </b><a NAME="E11:57"/>
 <a href="termord.html#K5">HM</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="termord.html#D8">TERMORD:def 8</a></i>;<br/>

<b>then </b> <a href="group_1.html#K2">1.</a> <font color="Maroon">L</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">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
<br/>.= 
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="termord.html#T22">TERMORD:22</a></i>
;<br/>
<a NAME="E13:57"/><b>then </b>
 <a href="termord.html#K4">HC</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E14:57"/>
<font color="Maroon">c</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/>


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