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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> <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="E1:65"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<a NAME="E2:65"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<a NAME="E3:65"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" title="UPROOTS:func.6">poly_quotient</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:65"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="uproots.html#D7" target="_self" title="UPROOTS:def.7">Def7</a></i>;<br/>
<b>assume </b><a NAME="E4:65"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" title="UPROOTS:func.6">poly_quotient</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<a NAME="E5:65"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" title="UPROOTS:func.6">poly_quotient</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E6:65"/><b>then </b>
 <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="uproots.html#E3:65"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="uproots.html#T48" target="_self" title="UPROOTS:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E7:65"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:65"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/>


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