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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="anproj_2.html#V6">Pappian</a> <a href="anproj_2.html#NM2">CollProjectivePlane</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <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>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E16</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>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>5</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>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E17</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E7:18"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E8:18"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</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>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</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>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</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>8</sub></font> )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1"/>
( not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><i><font color="Green">E25</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>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</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="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a></i>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>thus </b><a NAME="E2:18_2_1"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>thus </b><a NAME="E2:18_2_2"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E10:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_2_2_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_2_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E10:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E10:18"><i><font color="Green">E25</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_2_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:18_2"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3"/><i><font color="Green">E27</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>assume </b><a NAME="E2:18_3"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E3:18_3"/><i><font color="Green">E29</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>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</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="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:18_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:18_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:18_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:18_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:18_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:18_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:18_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:18_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><a NAME="E20:18_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>4</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>6</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>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_1_1"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_3_1"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3_1"/>
( <font color="Maroon">c<sub>4</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>6</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>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <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> )
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:18_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E22:18_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_2"/>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_2"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:18_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</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>17</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>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E2:18_3_3"/><i><font color="Green">E43</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>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E18</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E3:18_3_3"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_3"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:18_3_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_3_1"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:18_3_3_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_3_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:18_3_3_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:18_3_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:18_3_3"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:18_3_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:18_3_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E9:18_3_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</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>17</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>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="ref" href="hessenbe.html#T12" target="_self">Th12</a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:18_3"/>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E25:18_3"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E26:18_3"/><b>then </b><i><font color="Green">E42</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E27:18_3"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E28:18_3"/><b>then </b><i><font color="Green">E43</font></i>: 
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E29:18_3"/><b>then </b><i><font color="Green">E44</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E30:18_3"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E31:18_3"/><i><font color="Green">E46</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>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_4"/><i><font color="Green">E47</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>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_4_1"/>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_4_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:18_3_4_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_4_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E5:18_3_4_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_3_4"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_4"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:18_3_4"/>
( <font color="Maroon">c<sub>3</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>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:18_3"/>
( not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E33:18_3"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E34:18_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E35:18_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_5"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_5"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_5"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_5"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_5_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_5_1"/>
<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>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E18</font></i></a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E3:18_3_5_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3_5_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:18_3_5_1"/>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_3_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E15:18_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_5"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3_5"><i><font color="Green">E50</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E5:18_3_5"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:18_3_5"/>
( not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:18_3_5"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:18_3_5"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:18_3_5"/>
( not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:18_3_5"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E11:18_3_5"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:18_3_5"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E13:18_3_5"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E14:18_3_5"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="hessenbe.html#E15:18_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3_5"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hessenbe.html#E8:18_3_5"><i><font color="Green">E52</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E15:18_3_5"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:18_3"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_6"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_6"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E30:18_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_6"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="hessenbe.html#E35:18_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E37:18_3"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_7"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_7"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_7"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3_7"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E38:18_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_8"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_8"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_8"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_8"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E37:18_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_8"/>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:18_3_8"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E29:18_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3_8"><i><font color="Green">E53</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_8"><i><font color="Green">E54</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E39:18_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_9"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:18_3_9"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_9"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_9"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E38:18_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:18_3_9"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:18_3_9"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_9"><i><font color="Green">E53</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:18_3_9"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E38:18_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3_9"><i><font color="Green">E56</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:18_3_9"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E9:18_3_9"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_9"><i><font color="Green">E53</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E40:18_3"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_10"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_10"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E30:18_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_10"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:18_3_10"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:18_3_10"/>
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E39:18_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_10"><i><font color="Green">E54</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E41:18_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_11"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_11"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hessenbe.html#E35:18_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_11"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:18_3"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E40:18_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E43:18_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_12"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:18_3_12"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_12"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:18_3_12"/>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_12"><i><font color="Green">E57</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:18_3_12"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:18_3_12"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:18_3_12"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_12"><i><font color="Green">E57</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_12"><i><font color="Green">E58</font></i></a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E44:18_3"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_13"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_13"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_13"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E45:18_3"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_14_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:18_3_14_1"/><b>then </b><i><font color="Green">E59</font></i>: 
not <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_14_1"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:18_3_14_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:18_3_14_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:18_3_14_1"/><b>then </b>
not <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18"><i><font color="Green">E22</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:18_3_14_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:18_3_14_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E10:18_3_14_1"/><b>then </b><i><font color="Green">E60</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:18_3_14_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E12:18_3_14_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hessenbe.html#E10:18_3_14_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E13:18_3_14_1"/>
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_3_14"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_14_2"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:18_3_14_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_14_2"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:18_3_14_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3_14_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18_3_14"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_3"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_3"/>
( not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14"><i><font color="Green">E58</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_14_3"/><b>then </b>
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E43:18_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:18_3_14_3"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:18_3_14_3"/>
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:18_3_14_3"/><b>then </b>
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E2:18_3_14"><i><font color="Green">E59</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:18_3_14_3"/>
not <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_3_14"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="hessenbe.html#E41:18_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:18_3_14"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_4"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E2:18_3_14_4"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3_14"><i><font color="Green">E58</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_14_4"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:18_3_14_4"/><b>then </b>
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:18_3_14_4"/><b>then </b>
not <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_14"><i><font color="Green">E61</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:18_3_14_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3_14_5"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:18_3_14_5"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D14">ANPROJ_2:def 14</a></i>;<br/><a NAME="E4:18_3_14_5"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E5:18_3_14_5"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E26:18_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:18_3_14_5"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_5_1"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_14_5_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3_14_5_1"/>
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:18_3_14_5"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_5_2"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E2:18_3_14_5_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:18_3_14_5_2"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hessenbe.html#E35:18_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14_5"><i><font color="Green">E65</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_3_14_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E31:18_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3_14_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="hessenbe.html#E6:18_3_14_5"><i><font color="Green">E67</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E9:18_3_14_5"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:18_3_14_5"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E11:18_3_14_5"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E12:18_3_14_5"/>
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E13:18_3_14_5"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hessenbe.html#E22:18_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14"><i><font color="Green">E58</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3_14_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="hessenbe.html#E11:18_3_14_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E14:18_3_14_5"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E15:18_3_14_5"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E16:18_3_14_5"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E14:18_3_14_5"><i><font color="Green">E71</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E17:18_3_14_5"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E18:18_3_14_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3_14_5"><i><font color="Green">E70</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E19:18_3_14_5"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3_14_6"/><i><font color="Green">E64</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>15</sub></font>
 ;<br/><a NAME="E2:18_3_14_6"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E31:18_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="hessenbe.html#E36:18_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hessenbe.html#E40:18_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="hessenbe.html#E41:18_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><a NAME="E3:18_3_14_6"/><i><font color="Green">E66</font></i>: 
not <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_6_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E2:18_3_14_6_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:18_3_14_6_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E28:18_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hessenbe.html#E1:18_3_14_6"><i><font color="Green">E64</font></i></a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:18_3_14_6_1"/>
not <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:18_3_14_6"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_6_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_14_6_2"/>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:18_3_14_6"/>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_6_3"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_6_3"/>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_14_6_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_14_6_3_1"/>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hessenbe.html#E35:18_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:18_3_14_6_3_1"/>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_3_14_6_3"/>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E17:18_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:18_3_14_6"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18_3_14_6"><i><font color="Green">E65</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3_14_6"><i><font color="Green">E66</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_14_6"><i><font color="Green">E67</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E7:18_3_14_6"/>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E9:18_3_14_6"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:18_3_14_6"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E6:18_3_14_6"><i><font color="Green">E68</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E11:18_3_14_6"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E12:18_3_14_6"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18_3_14"><i><font color="Green">E58</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E13:18_3_14_6"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E17:18_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hessenbe.html#E34:18_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3_14_6"><i><font color="Green">E69</font></i></a>, <a class="txt" href="hessenbe.html#E10:18_3_14_6"><i><font color="Green">E70</font></i></a>, <a class="txt" href="hessenbe.html#E11:18_3_14_6"><i><font color="Green">E71</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E14:18_3_14_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E15:18_3_14_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E16:18_3_14_6"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E17:18_3_14_6"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E18:18_3_14_6"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E19:18_3_14_6"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E20:18_3_14_6"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E21:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hessenbe.html#E19:18_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="hessenbe.html#E31:18_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="hessenbe.html#E41:18_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="hessenbe.html#E42:18_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="hessenbe.html#E44:18_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="hessenbe.html#E4:18_3_14"><i><font color="Green">E61</font></i></a>, <a class="txt" href="hessenbe.html#E14:18_3_14_6"><i><font color="Green">E73</font></i></a>, <a class="txt" href="hessenbe.html#E15:18_3_14_6"><i><font color="Green">E74</font></i></a>, <a class="txt" href="hessenbe.html#E17:18_3_14_6"><i><font color="Green">E75</font></i></a>, <a class="txt" href="hessenbe.html#E19:18_3_14_6"><i><font color="Green">E76</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E22:18_3_14_6"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E23:18_3_14_6"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E22:18_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E24:18_3_14_6"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E18</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E25:18_3_14_6"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E26:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="hessenbe.html#E19:18_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="hessenbe.html#E20:18_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hessenbe.html#E21:18_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hessenbe.html#E23:18_3_14_6"><i><font color="Green">E77</font></i></a>, <a class="txt" href="hessenbe.html#E24:18_3_14_6"><i><font color="Green">E78</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E27:18_3_14_6"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E13:18_3_14_6"><i><font color="Green">E72</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E28:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E5:18_3_14"><i><font color="Green">E62</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E29:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E30:18_3_14_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hessenbe.html#E9:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="hessenbe.html#E3:18_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E31:18_3_14_6"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E2:18_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_3_14"/>
contradiction
 <b>by </b><i><a class="txt" href="hessenbe.html#E6:18_3_14"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:18"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E11:18"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
