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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#V7">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">F</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><br/> for <font color="Olive">F</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">p</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Olive">F</font>,<font color="Olive">i</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><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">F</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E2:18_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</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="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><a NAME="E3:18_1"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/><a NAME="E4:18_1"/><b>then </b>
<font color="Maroon">p</font> <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#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E5:18_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>;<br/><a NAME="E6:18_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E7:18_1"/><b>then </b>
 <a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon">p</font> <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> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><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">F</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM7">FinSequence</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="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">pg</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">rf</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E6:18_2_1"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/><a NAME="E7:18_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:18_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, </i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, </i>
;<br/><b>then </b><i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_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="E1:18_2_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><i><font color="Green">E45</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</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> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</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">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_3_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> or  not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2_1"><b>suppose </b></a><a NAME="E1:18_2_1_3_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:18_2_1_3_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:18_2_1_3_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E7:18_2_1_3_2_1"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="txt" href="hurwitz.html#E1:18"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_3_2_2"><b>suppose </b></a><a NAME="E1:18_2_1_3_2_2"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:18_2_1_3_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_2_2_1"><b>proof </b></a><div class="add">

 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</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> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</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">F</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:18_2_1_3_2_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( 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> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_3_2_2_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:18_2_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:18_2_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:18_2_1_3_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:18"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_3_2_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:18_2_1_3_2_2"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:18_2_1_3_2_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:18_2_1_3_2_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:18_2_1_3_2_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E9:18_2_1_3_2_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:18_2_1_3_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:18_2_1_3_2_2"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:18"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E12:18_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon">pg</font> <a href="polynom3.html#K8">+</a> <font color="Maroon">rf</font>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/><b>hence </b><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">pg</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</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="polynom3.html#D6">POLYNOM3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">rf</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:18"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:18"/><i><font color="Green">E99</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>
<b>thus </b><a NAME="E8:18"/>
<font color="Maroon">p</font> <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> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">F</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/>


</div>
