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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="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E50</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:63"/>
 <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:63"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:63"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="uproots.html#E1:63"><i><font color="Green">E50</font></i></a>, <a class="ref" href="uproots.html#T21" target="_self">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="uproots.html#E4:63"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E7:63"/>
<font color="Maroon">c<sub>4</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="txt" href="uproots.html#E4:63"><i><font color="Green">E51</font></i></a>, <a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E5:63"><i><font color="Green">E52</font></i></a>, <a class="ref" href="polynom5.html#T38">POLYNOM5:38</a></i>;<br/>


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