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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="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">p2</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> = <font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</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="25_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="E1:25_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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</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/><b>consider </b><font color="Maroon">rf</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:25_2"/><i><font color="Green">E30</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">rf</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">rf</font> holds <br/><font color="Maroon">rf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</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">p2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><b>consider </b><font color="Maroon">rp</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:25_2"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">rp</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">rp</font> holds <br/><font color="Maroon">rp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><i><font color="Green">E34</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">rp</font>
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">rp</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:25_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rf</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><i><font color="Green">E37</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">rf</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="hurwitz.html#E1:25"><i><font color="Green">E29</font></i></a>, , </i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">rf</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E10:25_2"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">rp</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rf</font> )
 ;<br/><a NAME="E2:25_2_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">rf</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:25_2_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">rp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="hurwitz.html#E6:25_2"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E7:25_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hurwitz.html#E6:25_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">rp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E7:25_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hurwitz.html#E6:25_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="hurwitz.html#E6:25_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hurwitz.html#E7:25_2"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</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">p2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</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">rf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E7:25_2"><i><font color="Green">E34</font></i></a></i>
;<br/></div><b>end;</b></div><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">rp</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">rp</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="hurwitz.html#E4:25_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>
;<br/><b>hence </b><a NAME="E13:25_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</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="E3:25"/>
<font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p2</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"><font color="Maroon">p1</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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