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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">r</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/>

<b>set </b><font color="Maroon">PRL</font> =  <a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font>;<br/>
<b>reconsider </b><font color="Maroon">pc</font> = <font color="Maroon">p</font>, <font color="Maroon">qc</font> = <font color="Maroon">q</font>, <font color="Maroon">rc</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>assume </b><a NAME="E3:43"/>
<font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E4:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
<font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:43"/><b>then </b>
<font color="Maroon">q</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">r</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon">q</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


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