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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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/>

<b>assume </b><a NAME="E2:80"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:80"><i><font color="Green">E72</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:80"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:80"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:80"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E10:80"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:80"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E13:80"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="txt" href="euclidlp.html#E10:80"><i><font color="Green">E77</font></i></a>, <a class="txt" href="euclidlp.html#E12:80"><i><font color="Green">E78</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:80"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:80"><i><font color="Green">E77</font></i></a>, <a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E16:80"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="txt" href="euclidlp.html#E10:80"><i><font color="Green">E77</font></i></a>, <a class="txt" href="euclidlp.html#E15:80"><i><font color="Green">E80</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E17:80"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E12:80"><i><font color="Green">E78</font></i></a>, <a class="txt" href="euclidlp.html#E15:80"><i><font color="Green">E80</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E19:80"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E12:80"><i><font color="Green">E78</font></i></a>, <a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E20:80"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:80"><i><font color="Green">E76</font></i></a>, <a class="txt" href="euclidlp.html#E17:80"><i><font color="Green">E82</font></i></a>, <a class="txt" href="euclidlp.html#E19:80"><i><font color="Green">E83</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:80"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E15:80"><i><font color="Green">E80</font></i></a>, <a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E23:80"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:80"><i><font color="Green">E76</font></i></a>, <a class="txt" href="euclidlp.html#E17:80"><i><font color="Green">E82</font></i></a>, <a class="txt" href="euclidlp.html#E22:80"><i><font color="Green">E85</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E24:80"/>
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E74</font></i></a>, <a class="txt" href="euclidlp.html#E20:80"><i><font color="Green">E84</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E25:80"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E23:80"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E26:80"/><i><font color="Green">E88</font></i>: 
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/>
<i><font color="Green">E90</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:80_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:80_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E25:80"><i><font color="Green">E87</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="txt" href="euclidlp.html#E4:80_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="euclidlp.html#E6:80_1"><i><font color="Green">E91</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<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>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
;<br/>
<b>hence </b><a NAME="E8:80_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:80"/><i><font color="Green">E89</font></i>: 
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:80_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>
<i><font color="Green">E91</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:80_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:80_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E25:80"><i><font color="Green">E87</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="txt" href="euclidlp.html#E4:80_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="euclidlp.html#E6:80_2"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<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>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
;<br/>
<b>hence </b><a NAME="E8:80_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E28:80"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E26:80"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E27:80"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="txt" href="euclidlp.html#E13:80"><i><font color="Green">E79</font></i></a>, <a class="txt" href="euclidlp.html#E16:80"><i><font color="Green">E81</font></i></a>, <a class="txt" href="euclidlp.html#E1:80"><i><font color="Green">E72</font></i></a></i>;<br/>


</div>
