<?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">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">p3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">q3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">r3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font>;<br/>



















<b>assume </b><a NAME="E1:16"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> &amp; not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">q3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><a NAME="E2:16_1_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:16_1_1"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E4:16_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E5:16_1_1"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:16_1_1"/>
<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:16_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:16_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q3</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E9:16_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
not <font color="Maroon">q3</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E10:16_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E11:16_1_1"/><b>then </b>
<font color="Maroon">q1</font>,<font color="Maroon">q3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E12:16_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">q3</font>,<font color="Maroon">q1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E13:16_1_1"/>
<font color="Maroon">q3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E14:16_1_1"/><b>then </b>
<font color="Maroon">r3</font>,<font color="Maroon">r2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E15:16_1_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><a NAME="E2:16_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_1_2_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_1_2_1"/><b>then </b>
not <font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E4:16_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_1_2_1"/>
( <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:16_1_2_1"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16_1_2_1"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:16_1_2_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E3:16_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_2"/>
( <font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_1_2_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_1_2_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1_2_2"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:16_1_2_2"/>
<font color="Maroon">q1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:16_1_2_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 ;<br/><a NAME="E2:16_1_3"/>
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_3_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_1_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E2:16_1_3_1_1"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_1_3_1_1"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:16_1_3_1_1"/><b>then </b>
( <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1_3_1_1"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16_1_3_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_1_3_1_1"/>
not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E8:16_1_3_1_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_1_3_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_1_3_1_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_1_3_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1_3_1_2"/>
( not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:16_1_3_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_3"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font>
 ;<br/><a NAME="E2:16_2_1_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_1_1_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_2_1_1_1_1"/>
not <font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:16_2_1_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_1_1_1"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_1_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_2_1_1_1_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_2_1_1_1_2"/>
not <font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:16_2_1_1_1_2"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_1_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font>
 ;<br/><a NAME="E2:16_2_1_2"/>
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1_2_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_2_1_2_1_1"/>
( <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_1_2_1_1"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:16_2_1_2_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_2_1_2_1_1"/>
not <font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_1_1"/><i><font color="Green">E85</font></i>: 
not <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_2_1_2_1_1_1"/>
<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_1_2_1_1_1"/><b>then </b>
not <font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_2_1_1_1"/>
not <font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_2_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_1_2_1"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E2:16_2_1_2_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2_1_2_1_2"/>
not <font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_2_1_2_1"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:16_2_1_2_1_2_1"/>
not <font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_2_1_2_1_2"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_1_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_1_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font> )
 ;<br/><a NAME="E2:16_2_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_1"/>
<font color="Maroon">p2</font>,<font color="Maroon">q3</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_2_1_3_1"/><b>then </b>
( <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E3:16_2_1_3_1"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">r1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_2_1_3_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_2_1_3_1"/>
not <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_1_1"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_2_1_3_1_1"/>
<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_2_1_3_1_1"/><b>then </b>
not <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_3_1_1"/>
not <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_3_1"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E3:16_2_1_3"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_3_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:16_2_1_3_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_1_3_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">r2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:16_2_1_3_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">p3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_2_1_3_2"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16_2_1_3_2"/><b>then </b>
not <font color="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:16_2_1_3_2"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_1_3"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2"/><i><font color="Green">E92</font></i>: 
not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:16_2_2_1_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_1_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">q3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2_2_1_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font>
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E4:16_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">r2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:16_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:16_2_2_1_1_1_1"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16_2_2_1_1_1_1"/><b>then </b>
( not <font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">p3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:16_2_2_1_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_1_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">r3</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:16_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_2_2_1_1_1_2"/>
not <font color="Maroon">q1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:16_2_2_1_1_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:16_2_2_1_2"/>
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_2_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_2_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_2_1_1"/>
( <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">r3</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_2_2_1_2_1_1"/>
not <font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:16_2_2_1_2_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r3</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_2_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_2_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font>
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/>
<a NAME="E2:16_2_2_1_2_1_2"/>
( <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">q3</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_2_2_1_2_1_2"/><b>then </b>
<font color="Maroon">p2</font>,<font color="Maroon">r1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:16_2_2_1_2_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q2</font>,<font color="Maroon">p3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16_2_2_1_2_1_2"/>
( not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:16_2_2_1_2_1_2"/><b>then </b>
not <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:16_2_2_1_2_1_2"/><b>then </b>
not <font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:16_2_2_1_2_1_2"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_1_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><a NAME="E2:16_2_2_1_3"/>
( <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2_1_3_1"/>
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_3_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q3</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_2_2_1_3_1_1"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2_2_1_3_1_1"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">r2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_2_2_1_3_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">q3</font>,<font color="Maroon">p3</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Maroon">p1</font>,<font color="Maroon">r2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_2_2_1_3_1_1"/>
not <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:16_2_2_1_3_1_1"/><b>then </b>
not <font color="Maroon">q3</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:16_2_2_1_3_1_1"/>
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_2_2_1_3_1"/>
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_1_3_1_2"/>
( <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon">q2</font>,<font color="Maroon">q3</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_2_2_1_3_1_2"/><b>then </b>
<font color="Maroon">p3</font>,<font color="Maroon">r1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:16_2_2_1_3_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">q3</font>,<font color="Maroon">p3</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Maroon">p2</font>,<font color="Maroon">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16_2_2_1_3_1_2"/>
( not <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:16_2_2_1_3_1_2"/><b>then </b>
not <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:16_2_2_1_3_1_2"/><b>then </b>
not <font color="Maroon">q3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:16_2_2_1_3_1_2"/>
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_2_2_1_3"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_2_1"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_2_2"/><i><font color="Green">E110</font></i>: 
( not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>consider </b><font color="Maroon">o</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E3:16_2_2_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">o</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="E4:16_2_2_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_2_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:16_2_2_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>

<a NAME="E3:16_2_2_2_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:16_2_2_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:16_2_2_2_1"/>
( <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E5:16_2_2_2"/>
not <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_2_2_2"/>
not <font color="Maroon">p1</font>,<font color="Maroon">o</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:16_2_2_2_2"/>
not <font color="Maroon">o</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , , , , , <a class="ref" href="anproj_2.html#D13">ANPROJ_2:def 13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:16_2"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">r3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div>
