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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <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>set </b><font color="Maroon">c<sub>5</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>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>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="E2:77"/><i><font color="Green">E62</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>3</sub></font> <a href="polynom3.html#K15">*'</a> <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>2</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="E3:77"/><i><font color="Green">E63</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</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/>
<b>consider </b><font color="Maroon">c<sub>7</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="E5:77"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>7</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>3</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>2</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="E6:77"/><i><font color="Green">E65</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#D8" target="_self">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E8:77"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</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>2</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="E9:77"/><i><font color="Green">E67</font></i>: 
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E10:77"/>
 <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>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:77"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>9</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="E13:77"/><i><font color="Green">E69</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>3</sub></font> <a href="polynom3.html#K15">*'</a> <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>2</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>9</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#E2:77"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E15:77"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom3.html#K15">*'</a> <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>2</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>9</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E13:77"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E16:77"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</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>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:77"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E19:77"/><i><font color="Green">E72</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>3</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>2</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>11</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#E5:77"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</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="E21:77"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</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>2</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>11</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E19:77"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E22:77"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</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>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E24:77"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E25:77"/><i><font color="Green">E75</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>2</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>13</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#E8:77"><i><font color="Green">E66</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</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="E27:77"/><i><font color="Green">E76</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>2</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>13</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E25:77"><i><font color="Green">E75</font></i></a></i>;<br/>
<i><font color="Green">E77</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>2</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>11</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom3.html#K15">*'</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>2</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>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E21:77"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uproots.html#E27:77"><i><font color="Green">E76</font></i></a>, <a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>2</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>11</sub></font></span>)</span> <a href="polynom3.html#K15">*'</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>2</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>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
<br/>.= 
<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>2</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="uproots.html#T32" target="_self">Th32</a></i>
<br/>.= 
<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>2</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
;<br/>
<a NAME="E29:77"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:77"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E30:77"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E12:77"><i><font color="Green">E68</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:77_2"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E3:77_2"/><b>then </b><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:77_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E3:77_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E7:77_2"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></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> <font color="Maroon">c<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> <a href="normsp_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</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="polynom3.html#T28">POLYNOM3:28</a>, <a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>;<br/><a NAME="E8:77_2"/><b>then </b>
<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>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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</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="E9:77_2"/><b>then </b><i><font color="Green">E81</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<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> <a href="polynom5.html#K2">`^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="uproots.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E10:77_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/><b>then </b><font color="Maroon">c<sub>3</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>4</sub></font> = 
<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>2</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</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>2</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>15</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E15:77"><i><font color="Green">E70</font></i></a>, <a class="ref" href="uproots.html#T32" target="_self">Th32</a></i>
<br/>.= 
<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>2</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <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>2</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>15</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
;<br/><a NAME="E12:77_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<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>2</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>15</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E28:77"><i><font color="Green">E77</font></i></a>, <a class="txt" href="uproots.html#E9:77_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="uproots.html#T30" target="_self">Th30</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>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> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>15</sub></font> = 
<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>2</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> 1</span>)</span> <a href="polynom3.html#K15">*'</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>2</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>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E6:77_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="uproots.html#T32" target="_self">Th32</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>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> <a href="polynom3.html#K15">*'</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>2</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>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>
;<br/><a NAME="E14:77_2"/><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>2</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>15</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font> <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> <font color="Maroon">c<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> <a href="polynom3.html#K15">*'</a> <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>2</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>16</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>;<br/><a NAME="E15:77_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E12:77_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="uproots.html#T51" target="_self">Th51</a></i>;<br/><a NAME="E16:77_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/><a NAME="E17:77_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_2_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>14</sub></font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E17:77_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1"><b>suppose </b></a><a NAME="E1:77_2_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:77_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/><a NAME="E3:77_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <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> <font color="Maroon">c<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> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T52" target="_self">Th52</a></i>;<br/><b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<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>2</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>11</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <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>2</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>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E21:77"><i><font color="Green">E73</font></i></a>, <a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
<br/>.= 
<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>2</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>11</sub></font></span>)</span> <a href="polynom3.html#K15">*'</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>2</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> 1</span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>
<br/>.= 
<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>2</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T32" target="_self">Th32</a></i>
;<br/><a NAME="E5:77_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:77"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E6:77_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E18:77"><i><font color="Green">E71</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E7:77_2_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_2"><b>suppose </b></a><a NAME="E1:77_2_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><a NAME="E2:77_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/><a NAME="E3:77_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <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> <font color="Maroon">c<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> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T52" target="_self">Th52</a></i>;<br/><b>then </b><font color="Maroon">c<sub>4</sub></font> = 
<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>2</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>13</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <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>2</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>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E27:77"><i><font color="Green">E76</font></i></a>, <a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
<br/>.= 
<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>2</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>13</sub></font></span>)</span> <a href="polynom3.html#K15">*'</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>2</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> 1</span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>
<br/>.= 
<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>2</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T32" target="_self">Th32</a></i>
;<br/><a NAME="E5:77_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E8:77"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E6:77_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E24:77"><i><font color="Green">E74</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E7:77_2_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E32:77"/>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:77"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E6:77"><i><font color="Green">E65</font></i></a>, <a class="txt" href="uproots.html#E9:77"><i><font color="Green">E67</font></i></a>, <a class="txt" href="uproots.html#E12:77"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uproots.html#E18:77"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uproots.html#E24:77"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uproots.html#E30:77"><i><font color="Green">E78</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div>
