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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D6" title="ALGSEQ_1:def.6">ALGSEQ_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D7" title="POLYNOM5:def.7">POLYNOM5:def 7</a></i>;<br/> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E3:36_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="hurwitz.html#E2:36"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="polynom5.html#T38" title="POLYNOM5:th.38">POLYNOM5:38</a></i>
;<br/><b>hence </b><a NAME="E5:36_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:36"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="hurwitz.html#E2:36"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="algseq_1.html#T31" title="ALGSEQ_1:th.31">ALGSEQ_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
not <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 ;<br/>


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