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<b>let </b><font color="Maroon">L</font> be  non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#NM1">comRing</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<a NAME="E1:74"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</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/>
<a NAME="E2:74"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="polynom5.html#T41">POLYNOM5:41</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:74"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">q</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Olive">k</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">q</font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:74"/><i><font color="Green">E44</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:74"/>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 1
 
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>;<br/>
<a NAME="E7:74"/><b>then </b>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 1</span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom3.html#T36">POLYNOM3:36</a></i>;<br/>
<a NAME="E8:74"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:74"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font>;<br/>
<a NAME="E10:74"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:74"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E13:74"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">q</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">k</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">q</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:74"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">k</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:74_1"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:74_1"/><b>then </b>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="polynom4.html#T5">POLYNOM4:5</a></i>;<br/><b>hence </b><a NAME="E4:74_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom4.html#T6">POLYNOM4:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:74"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E18:74"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> is <a href="uproots.html#V1" target="_self">non-zero</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><a NAME="E2:74_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:74_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K23">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:74_2"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="uproots.html#E2:74"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:74"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div>
