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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span> ;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span>  </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<font color="Maroon">p</font> = 
<font color="Maroon">p</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T5">POLYNOM4:5</a></i>
;<br/>
<a NAME="E2:48_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
;<br/>
<a NAME="E3:48_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E4:48_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 ex <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="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon">k</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="E6:48_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</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="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">f</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:48_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:48_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> &amp; 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">p</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/>
<a NAME="E12:48_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/>
<a NAME="E13:48_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E14:48_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:48_1_1"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <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="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:48_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">s</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="E17:48_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <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="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E18:48_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_3"/>
( <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> or <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_1"><b>suppose </b></a><a NAME="E1:48_1_1_3_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_3_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">s'</font> =  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="hurwitz.html#E2:48_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hurwitz.html#E1:48"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:48_1_1_3_1"/>
<font color="Maroon">s'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>hence </b><a NAME="E5:48_1_1_3_1"/>
 ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Olive">t</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, <a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_2"><b>suppose </b></a><a NAME="E1:48_1_1_3_2"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_3_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:48_1_1_3_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E4:48_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <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#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:48_1_1_3_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">f</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="E6:48_1_1_3_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <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="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">k'</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</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#T16">INT_1:16</a></i>;<br/><a NAME="E8:48_1_1_3_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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/><a NAME="E9:48_1_1_3_2"/><b>then </b><i><font color="Green">E45</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">f</font>
 ;<br/><a NAME="E10:48_1_1_3_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_3_2_1"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font>
 ;<br/><b>then reconsider </b><font color="Maroon">degn</font> = <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font></span>)</span> 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/><b>set </b><font color="Maroon">al</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>;<br/><b>set </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E3:48_1_1_3_2_1"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:48_1_1_3_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">degn</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1_3_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <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="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:48_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">degn</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_3_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><a NAME="E7:48_1_1_3_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:48_1_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E9:48_1_1_3_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3_2_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</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="48_1_1_3_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="E1:48_1_1_3_2_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_3_2_1_2_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:48_1_1_3_2_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">degn</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">degn</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E10:48_1_1_3_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T6" target="_self">Th6</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E4:48_1_1_3_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_3_2_1_2_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>
;<br/><b>hence </b><a NAME="E3:48_1_1_3_2_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><a NAME="E12:48_1_1_3_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">degn</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:48_1_1_3_2_1"/><b>then </b>
<font color="Maroon">degn</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>then </b><i><font color="Green">E93</font></i>: <font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">degn</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">degn</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E10:48_1_1_3_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E15:48_1_1_3_2_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font> holds <br/><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_4"><b>proof </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="E1:48_1_1_3_2_1_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font>
 ;<br/>

<a NAME="E2:48_1_1_3_2_1_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:48_1_1_3_2_1_4"/><b>then </b>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">degn</font></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E5:48_1_1_3_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E16:48_1_1_3_2_1"/>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">l</font> holds <br/><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_5"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">l</font> = <font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_5_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="E1:48_1_1_3_2_1_5_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:48_1_1_3_2_1_5_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_3_2_1_5_1"/>
<font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:48_1_1_3_2_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1_3_2_1_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 <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> holds <br/><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="algseq_1.html#D2">ALGSEQ_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">s1</font> = <font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>;<br/><a NAME="E18:48_1_1_3_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Olive">u</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> : 1 <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_6"><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="E1:48_1_1_3_2_1_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k'</font>
 ;<br/><i><font color="Green">E99</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K9">+</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">p</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T33">POLYNOM3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
 <a href="polynom3.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E137</font></i>: <font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E4:48_1_1_3_2_1_6"/><i><font color="Green">E138</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <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/><i><font color="Green">E139</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom3.html#D6">POLYNOM3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D7">POLYNOM3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<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">sf</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:48_1_1_3_2_1_6"/><i><font color="Green">E140</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sf</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">sf</font> holds <br/><font color="Maroon">sf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">m</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E8:48_1_1_3_2_1_6"/><i><font color="Green">E141</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">sf</font> <a href="hidden.html#R1">=</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">sf</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:48_1_1_3_2_1_6"/>
<font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E10:48_1_1_3_2_1_6"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_6_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3_2_1_6_4"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">sf</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:48_1_1_3_2_1_6_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sf</font> )
 <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></i>;<br/><a NAME="E3:48_1_1_3_2_1_6_4"/><b>then </b>
<font color="Maroon">m</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="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:48_1_1_3_2_1_6_4"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</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/><a NAME="E5:48_1_1_3_2_1_6_4"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:48_1_1_3_2_1_6_4"/>
<font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</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="txt" href="hurwitz.html#E4:48_1_1_3_2_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E145</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_2_1_6_5"><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="E1:48_1_1_3_2_1_6_5"/><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> <font color="Maroon">sf</font> &amp; <font color="Maroon">i'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>then </b><font color="Maroon">sf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i'</font></span>)</span></span>)</span>
<b>by </b><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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</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#T39">VECTSP_1:39</a></i>
;<br/><b>hence </b><a NAME="E3:48_1_1_3_2_1_6_5"/>
<font color="Maroon">sf</font> <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="hurwitz.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E13:48_1_1_3_2_1_6"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:48_1_1_3_2_1_6"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E15:48_1_1_3_2_1_6"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <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="E16:48_1_1_3_2_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E17:48_1_1_3_2_1_6"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">sf</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, , <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a></i>;<br/><b>then </b><i><font color="Green">E152</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sf</font> = 
<font color="Maroon">sf</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
<font color="Maroon">sf</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E14"><i><font color="Green">Lemma76</font></i></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">g1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E14"><i><font color="Green">Lemma76</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E3:48_1_1_3_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_2_1_6_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_3_2_1_6_7"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k'</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> )
 </span><b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</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="48_1_1_3_2_1_6_7_1"><b>suppose </b></a><a NAME="E1:48_1_1_3_2_1_6_7_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k'</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_3_2_1_6_7_1"/><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> <font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:48_1_1_3_2_1_6_7_1"/><b>then </b>
<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">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:48_1_1_3_2_1_6_7_1"/><b>then </b>
<font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><a NAME="E5:48_1_1_3_2_1_6_7_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">f</font> <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/><a NAME="E6:48_1_1_3_2_1_6_7_1"/><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font> 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="E8:48_1_1_3_2_1_6_7_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:48_1_1_3_2_1_6_7_1"/><b>then </b>
<font color="Maroon">e</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#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:48_1_1_3_2_1_6_7_1"/><b>then </b>
<font color="Maroon">e</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="E11:48_1_1_3_2_1_6_7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <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="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:48_1_1_3_2_1_6_7_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:48_1_1_3_2_1_6_7_1"/><b>then </b>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><a NAME="E14:48_1_1_3_2_1_6_7_1"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sf</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="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<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#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</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#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_2_1_6_7_2"><b>suppose </b></a><a NAME="E1:48_1_1_3_2_1_6_7_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font>
 ;<br/><div class="add"><i><font color="Green">E157</font></i>: <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">degn</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sf</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></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#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E20:48_1_1_3_2_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/><a NAME="E21:48_1_1_3_2_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k'</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/><a NAME="E22:48_1_1_3_2_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">g1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E23:48_1_1_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E9:48_1_1_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hurwitz.html#E5:48_1_1_3_2_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:48_1_1_3_2"/>
 ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Olive">t</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
