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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></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="E2:73"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <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">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:73"/><i><font color="Green">E59</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#D8" target="_self">Def8</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:73_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T52" target="_self">Th52</a></i>;<br/><a NAME="E3:73_1"/>
<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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 1 <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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>;<br/><a NAME="E4:73_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E2:73"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uproots.html#E2:73_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>hence </b><a NAME="E5:73_1"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E3:73"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:73"/><i><font color="Green">E60</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E6:73"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:73"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:73"/><i><font color="Green">E61</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:73"/><i><font color="Green">E62</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <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">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E2:73"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:73"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:73"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E12:73"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="uproots.html#E5:73"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:73"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E8:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> = 
<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">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E11:73"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E14:73"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom5.html#T20">POLYNOM5:20</a></i>
<br/>.= 
<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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
;<br/>
<b>hence </b><a NAME="E16:73"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#T51" target="_self">Th51</a></i>;<br/>


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