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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 ;<br/>

<a NAME="E2:66"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:66"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:66"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:66"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E9:66"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:66"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E12:66"/><i><font color="Green">E70</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/>1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_1"><i><font color="Green">E71</font></i></a></i>
;<br/><a NAME="E3:66_1"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><b>hence </b><a NAME="E4:66_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:66"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:66"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E16:66"/><i><font color="Green">E71</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E18:66"/><i><font color="Green">E72</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:66"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E20:66"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="jgraph_4.html#K7">-FanMorphE</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i><a class="ref" href="jgraph_4.html#T112">JGRAPH_4:112</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E21:66"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:66"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/>
<a NAME="E22:66"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E23:66"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/>
<a NAME="E24:66"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E25:66"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/>
<a NAME="E26:66"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E27:66"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/>
<a NAME="E28:66"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1"><b>case </b></a><a NAME="E1:66_2_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E3:66_2_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/>1 <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E27:66"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_1"><i><font color="Green">E86</font></i></a></i>
;<br/><a NAME="E3:66_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><a NAME="E4:66_2_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E5:66_2_1_1_1"/><i><font color="Green">E88</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:66_2_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/><a NAME="E7:66_2_1_1_1"/>
1 <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="ref" href="jgraph_2.html#T8">JGRAPH_2:8</a></i>;<br/><a NAME="E8:66_2_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then </b><i><font color="Green">E91</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:66_2_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E10:66_2_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E11:66_2_1_1_1"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:66_2_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E8:66_2_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E12:66_2_1_1_1"/><i><font color="Green">E94</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E10:66_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jgraph_4.html#T118">JGRAPH_4:118</a></i>;<br/><a NAME="E13:66_2_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E9:66_2_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/><b>then </b>1 <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:66_2_1_1_1"><i><font color="Green">E94</font></i></a></i>
;<br/><a NAME="E15:66_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><a NAME="E16:66_2_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E12:66_2_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E17:66_2_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T109">JGRAPH_4:109</a></i>;<br/><a NAME="E18:66_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:66_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:66_2_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66_2_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E17:66_2_1_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E20:66_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E5:66_2_1_1"/><i><font color="Green">E86</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E6:66_2_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1"><i><font color="Green">E89</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b>1 = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><a NAME="E3:66_2_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><a NAME="E2:66_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:66_2_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_1_1_1"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:66_2_1_1_2_1_1"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_2_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:66_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><a NAME="E3:66_2_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_2_1_1_2_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:66_2_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1_2_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><a NAME="E3:66_2_1_1_2_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_1_2_1_1_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E28:66"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_2_1_1_2_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_1_2_1_1_2_1_2"/><i><font color="Green">E98</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_1_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_1_2_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E28:66"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_1_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_1_2_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:66_2_1_1_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_1_2_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_1_2_1_2"/><i><font color="Green">E94</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_1_2_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:66_2_1_1_2_1_1_2_1_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:66"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E96</font></i></a></i>;<br/><a NAME="E4:66_2_1_1_2_1_1_2_1_2_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_1_2_1_2_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_2_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_1_2_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1_2_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><a NAME="E6:66_2_1_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E7:66_2_1_1_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><b>hence </b><a NAME="E8:66_2_1_1_2_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E28:66"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1_2_1_1_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:66_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_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="E1:66_2_1_1_2_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E6:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E7:66_2_1_1_2_1_2_1_1_1"/><i><font color="Green">E95</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E25:66"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/><a NAME="E8:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E9:66_2_1_1_2_1_2_1_1_1"/>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E27:66"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/><a NAME="E10:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E11:66_2_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E25:66"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_5.html#E27:66"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E12:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E13:66_2_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E14:66_2_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E8:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E10:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_5.html#E12:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_1_1_1_1"/><b>then </b>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_1_1_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_1_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_1_2_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_2"><i><font color="Green">E102</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_2_1_1_1_3_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E15:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E101</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_1_3_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_1_3_1_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_3_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_1_3_1_1"><i><font color="Green">E103</font></i></a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_2_1_1_1_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_3_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_1_3_1_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_1_3_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_1_3_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_5.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_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="E1:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_1_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_5.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_1_1_1_3_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E23:66"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jgraph_5.html#E24:66"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E25:66"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_1_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_1_1_1_3_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:66_2_1_1_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E28:66"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E14:66_2_1_1_2_1_2_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E16:66"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_1_1_2_1"/><b>then </b>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_1_1_2_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E12:66"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_2_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2_2"><i><font color="Green">E96</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_2_1_1_2_3_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_2_3_1_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_2_3_1_1"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2_3_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2_3_1_1"><i><font color="Green">E97</font></i></a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_2_1_2_1_1_2_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2_3_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2_3_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_2_3_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_2_3_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_5.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_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="E1:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2_3_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2_3_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jgraph_5.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_1_1_2_3_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E23:66"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jgraph_5.html#E24:66"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E25:66"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_1_1_2_3_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_1_1_2_3_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:66_2_1_1_2_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E26:66"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_2"/><b>then </b>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:66"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_2_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_3"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E12:66"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_3"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_1_2_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_3"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_4"/><b>then </b>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_2_4"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_5"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:66"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_5"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_5"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_5"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_1_2_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_5"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_2_6_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1_2_1_2"><i><font color="Green">E92</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_6_1_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_2_1_2_6_1_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1_2_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_1_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_2_1_2_6_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_1_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jgraph_5.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_2_1_2_6_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E22:66"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E24:66"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_6_1_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:66_2_1_1_2_1_2_6_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_5.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_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="E1:66_2_1_1_2_1_2_6_1_2_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1_2_1_2"><i><font color="Green">E94</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_2_1_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_2_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E6:66_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_1"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E4:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_2_1_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_1_2_1_2_6_1_2_1_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E7:66_2_1_1_2_1_2_6_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><a NAME="E4:66_2_1_1_2_1_2_6_1_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_2_1_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_5.html#T57" target="_self">Th57</a></i>;<br/><a NAME="E5:66_2_1_1_2_1_2_6_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_1_2_1_2_6_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E22:66"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_4.html#T89">JGRAPH_4:89</a>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_2_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_2_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_5.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_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="E1:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E22:66"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E7:66"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_5.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E3:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:66"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E9:66"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1_2_1_2_6_1_2_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_5.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_6_1_2_1_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E21:66"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jgraph_5.html#E22:66"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E23:66"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jgraph_5.html#E24:66"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2_6_1_2_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_5.html#T58" target="_self">Th58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_6_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1_1_2_1_2_6_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:66_2_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E18:66"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_5.html#E4:66_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66_2_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_4.html#T113">JGRAPH_4:113</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:66_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_4.html#T104">JGRAPH_4:104</a></i>;<br/><b>hence </b><a NAME="E9:66_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>6</sub></font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:66"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E7:66_2_1_1"><i><font color="Green">E88</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2"><b>case </b></a><a NAME="E1:66_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_2"/><i><font color="Green">E83</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><a NAME="E3:66_2_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E4:66_2_1_2"/><i><font color="Green">E85</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</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="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jgraph_5.html#T38" target="_self">Th38</a></i>;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:66_2_1_2_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:66_2_1_2"><i><font color="Green">E85</font></i></a></i>;<br/><b>thus </b><a NAME="E4:66_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_2_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:66_2_1_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:66_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jgraph_5.html#E3:66_2_1_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E8:66_2_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E9:66_2_1_2"/><b>then </b>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:66_2_1_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E12:66_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E8:66_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E13:66_2_1_2"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:66_2_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:66_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jgraph_5.html#E5:66_2_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><a NAME="E16:66_2_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:66_2_1_2_2"/>
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E2:66_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:66_2_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E18:66_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:66_2_1_2"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E19:66_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>6</sub></font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:66"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jgraph_5.html#E1:66_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E7:66_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_5.html#E11:66_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E15:66_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E17:66_2_1_2"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E30:66"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>6</sub></font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>


</div>
