<?xml version="1.0"?>
<div class="add">

<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="group_1.html#V2">unital</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">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<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="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><b>take </b><font color="Maroon">s</font> =  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>;<br/><b>thus </b><a NAME="E3:47_1_1"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:47_1_2"/>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E5:47_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T41">NAT_1:41</a></i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:47_1_2_1"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:47_1_2_1"/><b>then </b>
not <font color="Maroon">p</font> is <a href="polynom5.html#V1">with_roots</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:47_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E9:47_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom5.html#D7">POLYNOM5:def 7</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Olive">b</font>,<font color="Maroon">z</font></span>)</span> ) ) );<br/><a NAME="E7:47_1_2"/><i><font color="Green">E37</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>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of 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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2"><b>proof </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="E1:47_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 ;<br/>

<a NAME="E2:47_1_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:47_1_2_2"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:47_1_2_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of 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="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>thus </b><a NAME="E3:47_1_2_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:47_1_2_2_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of 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="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><a NAME="E3:47_1_2_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">t</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_2_2_1_2_1"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Olive">b</font>,<font color="Maroon">z</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
;<br/>

<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E1:47_1_2_2_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_2_2_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:47_1_2_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">t</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">hs</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><b> such that </b><br/><a NAME="E9:47_1_2"/><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">hs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</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="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>);<br/></i><a NAME="E10:47_1_2"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">hs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/><font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Olive">i</font>,<font color="Maroon">z</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_1_2_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">hs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E2:47_1_2_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_2_3"/>
<font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_3_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:47_1_2_3_1"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/><a NAME="E2:47_1_2_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:47_1_2_3_1"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Olive">b</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:47_1_2_3_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">b</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:47_1_2_3_1"/>
<font color="Maroon">hs</font> <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">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">i</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_2_3"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/><font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Olive">i</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Olive">b</font>,<font color="Maroon">z</font></span>)</span> );<br/><a NAME="E11:47_1_2"/><i><font color="Green">E62</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>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of 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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_4"><b>proof </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="E1:47_1_2_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of 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="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>take </b>
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
;<br/>

<b>take </b>
<font color="Maroon">k</font>
;<br/>

<a NAME="E3:47_1_2_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <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/>
<b>hence </b><a NAME="E4:47_1_2_4"/>
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>

<b>thus </b><a NAME="E5:47_1_2_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E6:47_1_2_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">h</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><b> such that </b><br/><a NAME="E13:47_1_2"/><i><font color="Green">E63</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</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="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E14:47_1_2"/><i><font color="Green">E90</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Olive">i</font>,<font color="Maroon">z</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_1_2_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</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="hs2(this)" href="javascript:()" title="47_1_2_5_1"><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="E1:47_1_2_5_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 ;<br/><a NAME="E2:47_1_2_5_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:47_1_2_5_1"/><i><font color="Green">E91</font></i>: 
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Olive">b</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:47_1_2_5_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">b</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a></i>;<br/><b>thus </b><a NAME="E7:47_1_2_5_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_2_5"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Olive">i</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">s</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">hs</font>;<br/><b>set </b><font color="Maroon">rs</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">h</font>;<br/><b>reconsider </b><font color="Maroon">s</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">hs</font>, <font color="Maroon">rs</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">h</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>take </b>
<font color="Maroon">s</font>
;<br/><a NAME="E16:47_1_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">rs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_6"><b>proof </b></a><div class="add">

<i><font color="Green">E94</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></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> <font color="Maroon">rs</font>
<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="47_1_2_6_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="E1:47_1_2_6_2"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">rs</font>
 ;<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>reconsider </b><font color="Maroon">r1z</font> =  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</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/><a NAME="E4:47_1_2_6_2"/><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_6_2_1"><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="E1:47_1_2_6_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 ;<br/><a NAME="E2:47_1_2_6_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:47_1_2_6_2_1"/><b>then </b><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_6_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_2_6_2_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:47_1_2_6_2_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">hs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">r1z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">hs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T36">POLYNOM3:36</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, , <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_2_6_2_1_1_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><i><font color="Green">E139</font></i>: <font color="Maroon">hs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">hs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="ref" href="hurwitz.html#T3" target="_self">Th3</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">r1z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">hs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K4" target="_self">qpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">k</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:47_1_2_6_2"/><b>then </b>
<font color="Maroon">r1z</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">hs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/><b>hence </b><a NAME="E7:47_1_2_6_2"/>
<font color="Maroon">rs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_2_6"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">rs</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E140</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_7"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:47_1_2_7"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 ;<br/><a NAME="E2:47_1_2_7"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:47_1_2_7"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:47_1_2_7"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E5:47_1_2_7"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:47_1_2_7"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <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">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T5" target="_self">Th5</a></i>;<br/><b>set </b><font color="Maroon">co</font> =  <a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font>;<br/><i><font color="Green">E145</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> = 
 <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">h</font></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> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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">h</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:47_1_2_7"/><b>then </b><i><font color="Green">E146</font></i>: 
<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="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></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="47_1_2_7_2"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:47_1_2_7_2"/><i><font color="Green">E147</font></i>: 
( <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="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">i'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 ;<br/><b>then consider </b><font color="Maroon">ci</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:47_1_2_7_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">ci</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ci</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E4:47_1_2_7_2"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:47_1_2_7_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">i'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><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> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i'</font>,<font color="Maroon">z</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#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="txt" href="hurwitz.html#E11:47_1_2"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i'</font>,<font color="Maroon">z</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="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E7:47_1_2_7_2"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</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="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E151</font></i>:  <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">h</font>,<font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>consider </b><font color="Maroon">cc</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:47_1_2_7"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cc</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E13:47_1_2_7"/>
<font color="Maroon">h</font> <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">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E14:47_1_2_7"/>
<font color="Maroon">rs</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i</font>,<font color="Maroon">z</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#E11:47_1_2"><i><font color="Green">E62</font></i></a>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_8"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:47_1_2_8"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><b>set </b><font color="Maroon">co</font> =  <a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font>;<br/><i><font color="Green">E155</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> = 
 <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">h</font></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> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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">h</font>,<font color="Maroon">i</font></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="47_1_2_8_2"><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="E1:47_1_2_8_2"/><i><font color="Green">E156</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">h</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">ci</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:47_1_2_8_2"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">ci</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</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">h</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">ci</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E4:47_1_2_8_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:47_1_2_8_2"/><b>then </b><i><font color="Green">E158</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">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:47_1_2_8_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">j</font>,<font color="Maroon">z</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#E5"><i><font color="Green">Lemma52</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">j</font>,<font color="Maroon">z</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="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="txt" href="hurwitz.html#E10:47_1_2"><i><font color="Green">E44</font></i></a>, , </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/></div><b>end;</b></div><a NAME="E4:47_1_2_8"/><b>then </b>
 <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">h</font>,<font color="Maroon">i</font></span>)</span> <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#T1">POLYNOM3:1</a></i>;<br/><b>hence </b><a NAME="E5:47_1_2_8"/>
<font color="Maroon">rs</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/>;<br/></div><b>end;</b></div><a NAME="E19:47_1_2"/>
<font color="Maroon">rs</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_9"><b>proof </b></a><div class="add">

<i><font color="Green">E172</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> = 
 <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> <font color="Maroon">rs</font>
<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="47_1_2_9_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="E1:47_1_2_9_2"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_9_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_1"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_1"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">evp</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:47_1_2_9_2_1_1"/><i><font color="Green">E175</font></i>: 
(  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">evp</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">evp</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">evp</font> holds <br/><font color="Maroon">evp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</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"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/><a NAME="E4:47_1_2_9_2_1_1"/><i><font color="Green">E176</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">evp</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="txt" href="hurwitz.html#E9:47_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><b>set </b><font color="Maroon">co</font> =  <a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0;<br/><i><font color="Green">E177</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> = 
 <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">h</font></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> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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">h</font>,0</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><i><font color="Green">E178</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_9_2_1_1_2"><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="E1:47_1_2_9_2_1_1_2"/><i><font color="Green">E179</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">h</font>,0</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">ci</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:47_1_2_9_2_1_1_2"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">ci</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</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">h</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ci</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, </i>;<br/><b>reconsider </b><font color="Maroon">aj</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">j</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E5:47_1_2_9_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E11:47_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:47_1_2_9_2_1_1_2"/><b>then </b><i><font color="Green">E182</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">m1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">j</font>,<font color="Maroon">z</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#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="txt" href="hurwitz.html#E13:47_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">j</font>,<font color="Maroon">z</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="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">aj</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="txt" href="hurwitz.html#E5:47_1_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T40">VECTSP_1:40</a></i>
;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">cop</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span>;<br/><i><font color="Green">E184</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> = 
<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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></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">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</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">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><b>then </b><i><font color="Green">E185</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> = 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">evp</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, </i>
;<br/><b>then </b><i><font color="Green">E186</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></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">evp</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">evp</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="hs2(this)" href="javascript:()" title="47_1_2_9_2_1_1_6"><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="E1:47_1_2_9_2_1_1_6"/><i><font color="Green">E187</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"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span>
 ;<br/><a NAME="E2:47_1_2_9_2_1_1_6"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <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"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</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 NAME="E3:47_1_2_9_2_1_1_6"/><b>then </b><i><font color="Green">E189</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:47_1_2_9_2_1_1_6"/><i><font color="Green">E190</font></i>: 
 <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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></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/><i><font color="Green">E191</font></i>: <font color="Maroon">evp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<font color="Maroon">evp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</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"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_1_6_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_9_2_1_1_6_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_1_6_2_1"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_1_6_2_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:47_1_2_9_2_1_1_6_2_1"/><b>then </b>
<font color="Maroon">j</font> <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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></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">E193</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></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/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E4:47_1_2_9_2_1_1_6_2_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><font color="Maroon">evp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, , <a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><b>hence </b><a NAME="E6:47_1_2_9_2_1_1_6_2_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">evp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_1_6_2_2"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_1_6_2_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:47_1_2_9_2_1_1_6_2_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:47_1_2_9_2_1_1_6_2_2"/><b>then </b><i><font color="Green">E197</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:47_1_2_9_2_1_1_6_2_2"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:47_1_2_9_2_1_1_6_2_2"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <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">h</font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E11:47_1_2"><i><font color="Green">E62</font></i></a>, </i>;<br/><a NAME="E7:47_1_2_9_2_1_1_6_2_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">j1</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">h</font>,0</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:47_1_2_9_2_1_1_6_2_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="xxreal_0.html#NR3" target="_self">&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> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green">E199</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></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#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a></i>
;<br/><a NAME="E10:47_1_2_9_2_1_1_6_2_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:47_1_2_9_2_1_1_6_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E12:47_1_2_9_2_1_1_6_2_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">evp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , , <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><b>then </b> <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">evp</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E13:47_1_2"><i><font color="Green">E63</font></i></a>, </i>
<br/>.= 
<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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">h</font>,0</span>)</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#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/><a NAME="E12:47_1_2_9_2_1_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</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> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">h</font>,0</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/><b>then </b><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K1" target="_self">Coeff</a> <font color="Maroon">h</font>,0</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">h</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</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">h</font>,0</span>)</span>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E14:47_1_2_9_2_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_2"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_2_9_2_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_2_1_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">rs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="hurwitz.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
<b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2_9_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:47_1_2_9_2_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">rs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><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> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i</font>,<font color="Maroon">z</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="hurwitz.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> <font color="Maroon">i</font>,<font color="Maroon">z</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="polynom5.html#D3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="txt" href="hurwitz.html#E5:47_1_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_2_9"/>
<font color="Maroon">rs</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:47_1_2"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
