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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E44</font></i>: 
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="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:88"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:88"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:88"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">s</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:88_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:88_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:88_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E6:88_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:88_1"/><i><font color="Green">E53</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E8:88_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>set </b><font color="Maroon">ff</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font>;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Olive">n</font></span>)</span> );<br/><a NAME="E9:88_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><a NAME="E10:88_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>proof </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="E1:88_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> 
 ;<br/>

<b>then 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="E3:88_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:88_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 ;<br/>
<b>take </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>;<br/>

<b>thus </b><a NAME="E5:88_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:88_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> 
 <b>and </b><br/><a NAME="E13:88_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="ref" href="uproots.html#D1" target="_self">Def1</a>);<br/></i><a NAME="E14:88_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:88_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:88_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:88_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>consider </b><font color="Maroon">n1</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="E5:88_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E6:88_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E8:88_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E9:88_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E10:88_1_2"/>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:88_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:88_1_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:88_1_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<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="E6:88_1_3_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E7:88_1_3_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">n</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="E9:88_1_3_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E10:88_1_3_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:88_1_3_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:88_1_3_1"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> &amp; 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:88_1_3_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E14:88_1_3_1"/><b>then </b><i><font color="Green">E101</font></i>: 
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:88_1_3_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:88_1_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T33">POLYNOM1:33</a></i>;<br/><a NAME="E16:88_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E7:88_1_3_1"><i><font color="Green">E96</font></i></a>, </i>;<br/><a NAME="E17:88_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E18:88_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E9:88_1_3_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:88_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:88_1_3"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">yn</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E6:88_1_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:88_1_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:88_1_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">yn</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E9:88_1_3"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">yn</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/><a NAME="E10:88_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/><b>then consider </b><font color="Maroon">j1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:88_1_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j1</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E14:88_1_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E15:88_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E16:88_1_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:88_1_3"/>
( <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font> &amp; <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/><a NAME="E18:88_1_3"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T31">RLVECT_1:31</a></i>;<br/><a NAME="E19:88_1_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E20:88_1_3"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:88_1_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="uproots.html#T3" target="_self">Th3</a>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E22:88_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E23:88_1_3"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:88_1_3"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:88_1_3"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E27:88_1_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E28:88_1_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E29:88_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div><a NAME="E16:88_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E17:88_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i>, , <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><b>hence </b><a NAME="E18:88_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E8:88"/><i><font color="Green">E120</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:88"/><i><font color="Green">E121</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:88"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E12:88"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>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>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E15:88"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E16:88"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E17:88"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E18:88"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<a NAME="E19:88"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E20:88"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E21:88"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:88"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:88"/>
( <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c</font> &amp; <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/>
<a NAME="E24:88"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" target="_self">Def1</a>, , , <a class="ref" href="rlvect_1.html#T31">RLVECT_1:31</a></i>;<br/>
<a NAME="E25:88"/>
( <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E26:88"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div>
