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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#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#V4">right-distributive</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>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">f</font> =  <a href="polynom3.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span>;<br/>
<i><font color="Green">E29</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D7">POLYNOM3:def 7</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T40">VECTSP_1:40</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D7">POLYNOM3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom5.html#D3">POLYNOM5:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:22_2"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
 <a href="polynom3.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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