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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#V3" title="RLVECT_1:attr.3">Abelian</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_2.html#V2" title="VECTSP_2:attr.2">domRing-like</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>, <font color="Maroon" title="c4">r</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="E1:43"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">PRL</font> =  <a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">pc</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c7">qc</font> = <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c8">rc</font> = <font color="Maroon" title="c4">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<b>assume </b><a NAME="E3:43"/>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E4:43"/><b>then </b>
<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="polynom3.html#K11" title="POLYNOM3:func.11">-</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="c4">r</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#T30" title="POLYNOM3:th.30">POLYNOM3:30</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T28" target="_self" title="UPROOTS:th.28">Th28</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <font color="Maroon" title="c4">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:43"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="uproots.html#T23" target="_self" title="UPROOTS:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="uproots.html#T29" target="_self" title="UPROOTS:th.29">Th29</a></i>;<br/>


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