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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_2.html#NM1">comRing</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = 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">c<sub>1</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:91"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:91"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:91"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:91"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<a NAME="E5:91"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E6:91"/><b>then </b><i><font color="Green">E76</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:91"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E8:91"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E9:91"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T23">FINSEQ_2:23</a></i>;<br/>
<a NAME="E10:91"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E11:91"/><i><font color="Green">E80</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp;  <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E12:91"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> as    <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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:91_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uproots.html#E6:91"><i><font color="Green">E76</font></i></a>, <a class="ref" href="qc_lang4.html#T7">QC_LANG4:7</a></i>;<br/>
<a NAME="E5:91_1"/><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:91_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uproots.html#E6:91"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E8:91_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E5:91_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:91_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E11:91_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E4:91_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E14:91_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:91"><i><font color="Green">E76</font></i></a>, <a class="ref" href="qc_lang4.html#T7">QC_LANG4:7</a></i>;<br/>
<a NAME="E15:91_1"/><i><font color="Green">E87</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E6:91"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E17:91_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E5:91_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E18:91_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E14:91_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E19:91_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="uproots.html#E4:91_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="uproots.html#E8:91_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="uproots.html#E11:91_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>

<a NAME="E20:91_1"/><i><font color="Green">E90</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E14:91_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="fvsum_1.html#T99">FVSUM_1:99</a></i>;<br/>
<a NAME="E21:91_1"/>
<font color="Maroon">c<sub>15</sub></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<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:91"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uproots.html#E15:91_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:91_1"/>
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E17:91_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="uproots.html#E18:91_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="uproots.html#E20:91_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="uproots.html#T63" target="_self">Th63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:91"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:91_2"/><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:91_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>given </b><font color="Maroon">c<sub>10</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <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">c<sub>1</sub></font></span>)</span><b> such that </b><a NAME="E5:91_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:91_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:91_2"/><i><font color="Green">E87</font></i>: 
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E9:91_2"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="uproots.html#E1:91_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uproots.html#E2:91_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:91_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:91_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E13:91_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>;<br/>
<a NAME="E14:91_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>;<br/>
<a NAME="E15:91_2"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E10:91_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_7.html#T6">FUNCT_7:6</a></i>;<br/>
<a NAME="E16:91_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E17:91_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="uproots.html#E5:91_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="uproots.html#E14:91_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E19:91_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <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<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:91_2"/><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:91_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E26:91_2"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uproots.html#E25:91_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E27:91_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E28:91_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E29:91_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E6:91_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E30:91_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<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<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>
<b>by </b><i><a class="txt" href="uproots.html#E2:91"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uproots.html#E11:91"><i><font color="Green">E80</font></i></a>, <a class="txt" href="uproots.html#E10:91_2"><i><font color="Green">E89</font></i></a></i>
;<br/>
<a NAME="E32:91_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</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<sub>14</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="uproots.html#E26:91_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="uproots.html#E29:91_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E33:91_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="uproots.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E34:91_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_2_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:91_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:91_2_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:91_2_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="uproots.html#E4:91_2_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:91_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:91_2_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:91_2_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:91_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="uproots.html#E5:91_2_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:91_2_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E8:91_2_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E12:91_2_2"/><i><font color="Green">E103</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="uproots.html#E10:91_2_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:91_2_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:91_2_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E8:91_2_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="uproots.html#E9:91_2_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E15:91_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uproots.html#E10:91"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E9:91_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="uproots.html#E6:91_2_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="uproots.html#E12:91_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="uproots.html#E13:91_2_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="graph_5.html#T10">GRAPH_5:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:91_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a>  <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E17:91_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="uproots.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b> <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> = 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="uproots.html#T66" target="_self">Th66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E7:91_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="uproots.html#E33:91_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E32:91_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="fvsum_1.html#T100">FVSUM_1:100</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E14:91"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(<a class="txt" href="uproots.html#E12:91"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uproots.html#E13:91"><i><font color="Green">E82</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E16:91"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E17:91"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:91"/><i><font color="Green">E85</font></i>: 
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E6:91"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>thus </b><a NAME="E19:91"/>
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:91"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E3:91"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uproots.html#E7:91"><i><font color="Green">E77</font></i></a>, <a class="txt" href="uproots.html#E9:91"><i><font color="Green">E78</font></i></a>, <a class="txt" href="uproots.html#E10:91"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E16:91"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uproots.html#E17:91"><i><font color="Green">E84</font></i></a>, <a class="txt" href="uproots.html#E18:91"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_2.html#T23">FINSEQ_2:23</a></i>;<br/>


</div>
