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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E39">A2</font></i>: 
(  not <font color="Maroon" title="c2">p</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> or  not <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> )
 ;<br/>

<a NAME="E3:49"/>
(  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="uproots.html#E2:49"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<a NAME="E4:49"/><b>then </b>
(  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
;<br/>
<a NAME="E5:49"/><b>then </b>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="uproots.html#T35" target="_self" title="UPROOTS:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E6:49"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:49"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>


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