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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="polynom5.html#V2">algebraic-closed</a> <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:82"/><i><font color="Green">E66</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="uproots.html#T19" target="_self">Th19</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="algseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1;<br/>
<a NAME="E2:82"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:82_1"/><i><font color="Green">E69</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:82_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E4:82_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uproots.html#E3:82_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>thus </b><a NAME="E5:82_1"/>
 <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="uproots.html#E4:82_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b> <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
1 <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="uproots.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uproots.html#E1:82_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="polynom5.html#V1">with_roots</a>
 <b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="polynom5.html#D8">POLYNOM5:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:82_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="polynom5.html#D7">POLYNOM5:def 7</a></i>;<br/><a NAME="E5:82_1_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E4:82_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="uproots.html#T54" target="_self">Th54</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being  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 <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:82_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E8:82_1_1_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E9:82_1_1_1_2"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:82_1_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E12:82_1_1_1_2"/><i><font color="Green">E78</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:82_1_1_1_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:82_1_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E12:82_1_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</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>5</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>10</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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="uproots.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E16:82_1_1_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="uproots.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E17:82_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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/><b>thus </b><a NAME="E18:82_1_1_1_2"/>
 <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:82_1_1_1_2_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E2:82_1_1_1_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="algseq_1.html#K3">len</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>5</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> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="polynom5.html#T41">POLYNOM5:41</a></i>;<br/>
<a NAME="E3:82_1_1_1_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="polynom5.html#T24">POLYNOM5:24</a></i>;<br/>
<a NAME="E4:82_1_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E5:82_1_1_1_2_1"/><i><font color="Green">E84</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="uproots.html#E5:82_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uproots.html#E8:82_1_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uproots.html#E11:82_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="uproots.html#E3:82_1_1_1_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:82_1_1_1_2_1"/><i><font color="Green">E85</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="uproots.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:82_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="uproots.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E8:82_1_1_1_2_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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<sub>5</sub></font></span>)</span>,<span class="p4">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p0">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1_2_1_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="uproots.html#E17:82_1_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_2_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><i><font color="Green">E88</font></i>:  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E8:82_1_1_1_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="polynom4.html#T13">POLYNOM4:13</a></i>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:82_1_1_1_2_1_1_1"><i><font color="Green">E87</font></i></a></i>
;<br/><b>thus </b> <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E69"><i><font color="Green">Lemma63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="txt" href="uproots.html#E1:82_1_1_1_2_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="uproots.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="ref" href="uproots.html#T60" target="_self">Th60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uproots.html#E4:82_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uproots.html#E3:82_1_1_1_2_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="uproots.html#E2:82_1_1_1_2_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_2_1_1_2"/><b>then </b>
(  <a href="algseq_1.html#K3">len</a> <span class="p1">(<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>5</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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E5:82_1_1_1_2_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="uproots.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E3:82_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p2">(<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<sub>5</sub></font></span>)</span>,<span class="p4">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</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="binarith.html#K5">-'</a> 1 &amp;  <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 )
 <b>by </b><i><a class="txt" href="uproots.html#E1:82_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uproots.html#E3:82_1_1_1_2_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="uproots.html#E6:82_1_1_1_2_1"><i><font color="Green">E85</font></i></a></i>;<br/><b>thus </b> <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E69"><i><font color="Green">Lemma63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E2:82_1_1_1_2_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="uproots.html#E4:82_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uproots.html#E3:82_1_1_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="polynom5.html#T24">POLYNOM5:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E5:82_1_1_1_2_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p5">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:82_1_1_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p0">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p0">(<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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E14:82_1_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uproots.html#E8:82_1_1_1_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="polynom4.html#T13">POLYNOM4:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E2:82_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uproots.html#E1:82_1_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:82"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="uproots.html#E2:82"><i><font color="Green">E67</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:82"/>
 <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E1:82"><i><font color="Green">E66</font></i></a></i>;<br/>


</div>
