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

<b>let </b><font color="Maroon">PCPP</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">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">b3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">c2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> )
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">c1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E6:19"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> )
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E59</font></i>: 
( not <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">x</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">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">y</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">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E7:19_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">a</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">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E9:19_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">z</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>assume </b><a NAME="E10:19_1"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>thus </b><a NAME="E2:19_1_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_1_1"/>
( not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_1_1_1"/>
( not <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:19_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E2:19_1_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E3:19_1_1_1_2"/>
( not <font color="Maroon">o</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:19_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:19_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_1_2_1"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c3</font>,<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="txt" href="hessenbe.html#E2:19"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E4:19_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:19_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="hessenbe.html#E1:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a>, <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:19_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_2_2"/>
( not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">b1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_1_2_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:19_1_1_1_2"/>
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E8:19_1_1_1_2"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, <a class="txt" href="hessenbe.html#E2:19"><i><font color="Green">E35</font></i></a>, <a class="txt" href="hessenbe.html#E3:19"><i><font color="Green">E36</font></i></a>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a>, , , <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19_1_1_1"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_1_3"/>
( not <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E2:19_1_1_1_3"/><b>then </b>
not <font color="Maroon">a2</font>,<font color="Maroon">c1</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:19_1_1_1_3"/>
not <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b3</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:19_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">x</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="E6:19_1_1_1"/>
( not <font color="Maroon">a1</font>,<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font>,<font color="Maroon">o</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E7:19_1_1_1"/><b>then </b>
not <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:19_1_1_1"/><b>then </b>
not <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:19_1_1_1"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">c2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , , <a class="txt" href="hessenbe.html#E1:19"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E2:19"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:19_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E79</font></i>: 
(  not <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:19_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:19_1_2_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:19_1_2_1"/>
( not <font color="Maroon">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E5:19_1_2_1"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="hessenbe.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_2"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , , , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:19_1"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , , , </i>;<br/>


</div>
