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<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>let </b><font color="Maroon">x</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="E1:18"/><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="E2:18"/><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="E3:18"/><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="E4:18"/><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="E5:18"/><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>  )
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a2</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E7:18"/><i><font color="Green">E59</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> 
 ;<br/>

<a NAME="E8:18"/><i><font color="Green">E60</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/>
<a NAME="E9:18"/><i><font color="Green">E61</font></i>: 
not <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">b1</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">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>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:18_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>, </i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
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/>


</div><b>end;</b></div>
<a NAME="E10:18"/><i><font color="Green">E62</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/>
<div><i><font color="Green">E63</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">E64</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  or <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</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="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/><b>thus </b><a NAME="E2:18_2_1"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</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">E67</font></i>: 
<font color="Maroon">a1</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="18_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1_1_1"/>
<font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_2_1_1_1"/><b>then </b>
<font color="Maroon">a2</font>,<font color="Maroon">a1</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="E3:18_2_1_1_1"/>
<font color="Maroon">a1</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="E2:18_2_1_1"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c2</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">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:18_2_1_1_2"/>
<font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">c2</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="E3:18_2_1_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>, , , <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">E68</font></i>: 
<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>thus </b><a NAME="E2:18_2_2"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font>,<font color="Maroon">c3</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">E69</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</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">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_2_2_1_1"/><b>then </b>
<font color="Maroon">b2</font>,<font color="Maroon">b1</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="E3:18_2_2_1_1"/>
<font color="Maroon">b1</font>,<font color="Maroon">b2</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="E2:18_2_2_1"/>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c2</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">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:18_2_2_1_2"/>
<font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c2</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="E3:18_2_2_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>, , , <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">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>
<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">E70</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>assume </b><a NAME="E2:18_3"/><i><font color="Green">E71</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/><a NAME="E3:18_3"/><i><font color="Green">E72</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>, <a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</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="E5:18_3"/><i><font color="Green">E73</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>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E7:18_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">z</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">p</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">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E9:18_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">c3</font>,<font color="Maroon">q</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">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E11:18_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">b2</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">a3</font>,<font color="Maroon">r</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="E13:18_3"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">a2</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">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PCPP</font><b> such that </b><br/><a NAME="E15:18_3"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">a3</font>,<font color="Maroon">q'</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="E17:18_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">b1</font>,<font color="Maroon">r</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p</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>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="E19:18_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">b3</font>,<font color="Maroon">r</font>,<font color="Maroon">z''</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p</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/><a NAME="E20:18_3"/><i><font color="Green">E81</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="18_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_1"/><i><font color="Green">E82</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="18_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_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="E2:18_3_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="E2:18_3_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="E3:18_3_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="hessenbe.html#T6" target="_self">Th6</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">E83</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E22:18_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</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">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_3_2"/><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>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:18_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E23:18_3"/><i><font color="Green">E85</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="18_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_3"/><i><font color="Green">E86</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="E2:18_3_3"/><i><font color="Green">E87</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="E3:18_3_3"/>
( 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="E4:18_3_3"/><b>then </b><i><font color="Green">E88</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="E5:18_3_3"/><i><font color="Green">E89</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="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">E90</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="E2:18_3_3_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>, <a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E3:18_3_3_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#T10" target="_self">Th10</a>, <a class="ref" href="hessenbe.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E4:18_3_3_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="E5:18_3_3_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T12" target="_self">Th12</a>, <a class="txt" href="hessenbe.html#E5:18"><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">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/>
<a NAME="E7:18_3_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<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/>
<a NAME="E8:18_3_3"/>
<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="E9:18_3_3"/>
( <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="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a>, <a class="ref" href="hessenbe.html#T13" target="_self">Th13</a>, <a class="ref" href="hessenbe.html#T14" target="_self">Th14</a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E46</font></i></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">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E25:18_3"/><b>then </b>
not <font color="Maroon">a1</font>,<font color="Maroon">c3</font>,<font color="Maroon">o</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">E38</font></i></a>, </i>;<br/><a NAME="E26:18_3"/><b>then </b><i><font color="Green">E92</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E27:18_3"/>
<font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E28:18_3"/><b>then </b><i><font color="Green">E93</font></i>: 
not <font color="Maroon">b1</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E29:18_3"/><b>then </b><i><font color="Green">E94</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E30:18_3"/><i><font color="Green">E95</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a3</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">E34</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E31:18_3"/><i><font color="Green">E96</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">p</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</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">E97</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</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">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_3_4_1"/><b>then </b>
not <font color="Maroon">a1</font>,<font color="Maroon">z</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E3:18_3_4_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">z</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:18_3_4_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E5:18_3_4_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_3_4"/>
( <font color="Maroon">a2</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">a2</font>,<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:18_3_4"/><b>then </b>
not <font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_3_4"/>
( <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">p</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="hessenbe.html#T16" target="_self">Th16</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">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">o</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>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E33:18_3"/><b>then </b>
not <font color="Maroon">a2</font>,<font color="Maroon">c3</font>,<font color="Maroon">o</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">E38</font></i></a>, </i>;<br/><a NAME="E34:18_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 <b>by </b><i><a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E35:18_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">a</font>,<font color="Maroon">q</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">E100</font></i>: 
( <font color="Maroon">a3</font>,<font color="Maroon">z</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a3</font>,<font color="Maroon">a2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:18_3_5"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:18_3_5"/>
<font color="Maroon">a</font>,<font color="Maroon">z</font>,<font color="Maroon">c3</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">E102</font></i>: 
( <font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">a</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">E34</font></i></a>, </i>;<br/>
<a NAME="E2:18_3_5_1"/>
<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="E3:18_3_5_1"/><b>then </b>
<font color="Maroon">c3</font>,<font color="Maroon">a</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_3_5_1"/>
<font color="Maroon">a</font>,<font color="Maroon">z</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_3_5"/><b>then </b>
<font color="Maroon">c3</font>,<font color="Maroon">q'</font>,<font color="Maroon">p</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">E34</font></i></a>, <a class="txt" href="hessenbe.html#E2:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, <a class="ref" href="hessenbe.html#T19" target="_self">Th19</a>, , </i>;<br/>
<a NAME="E5:18_3_5"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">c3</font>,<font color="Maroon">q'</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:18_3_5"/>
( not <font color="Maroon">a2</font>,<font color="Maroon">a1</font>,<font color="Maroon">a3</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">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> )
 
<b>by </b><i>, , <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E7:18_3_5"/><b>then </b>
not <font color="Maroon">a2</font>,<font color="Maroon">c3</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:18_3_5"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:18_3_5"/>
( not <font color="Maroon">a1</font>,<font color="Maroon">a3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <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">a1</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#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E10:18_3_5"/><b>then </b>
not <font color="Maroon">a1</font>,<font color="Maroon">z</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:18_3_5"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">z</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:18_3_5"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E13:18_3_5"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:18_3_5"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">a3</font>,<font color="Maroon">q</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">E35</font></i></a>, , , </i>;<br/>
<b>hence </b><a NAME="E15:18_3_5"/>
<font color="Maroon">a3</font>,<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E36:18_3"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">a3</font>,<font color="Maroon">q</font>,<font color="Maroon">o</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">a3</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T14" target="_self">Th14</a>, </i>;<br/>
<b>hence </b><a NAME="E2:18_3_6"/>
not <font color="Maroon">a3</font>,<font color="Maroon">q</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E37:18_3"/><i><font color="Green">E106</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</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">a3</font>,<font color="Maroon">b3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_3_7"/><b>then </b>
not <font color="Maroon">b2</font>,<font color="Maroon">o</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="E3:18_3_7"/>
not <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E38:18_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b2</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="18_3_8"><b>proof </b></a><div class="add">

<a NAME="E1:18_3_8"/><i><font color="Green">E108</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>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:18_3_8"/><i><font color="Green">E109</font></i>: 
( <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="E3:18_3_8"/><i><font color="Green">E110</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">b3</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18_3_8"/>
( <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="E5:18_3_8"/>
( <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c3</font> )
 <b>by </b><i>, , , <a class="ref" href="hessenbe.html#T13" target="_self">Th13</a>, , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E39:18_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</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">E112</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:18_3_9"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c1</font>,<font color="Maroon">b2</font>,<font color="Maroon">c3</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">E34</font></i></a>, </i>;<br/>
<a NAME="E3:18_3_9"/>
<font color="Maroon">c1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18_3_9"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:18_3_9"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c3</font>,<font color="Maroon">b2</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:18_3_9"/>
<font color="Maroon">c3</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</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">E34</font></i></a>, , </i>;<br/>
<a NAME="E7:18_3_9"/><b>then </b>
<font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:18_3_9"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:18_3_9"/>
contradiction
 <b>by </b><i>, <a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</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">E120</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</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">E121</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E2:18_3_10"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b2</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">E34</font></i></a>, </i>;<br/>
<a NAME="E3:18_3_10"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:18_3_10"/>
not <font color="Maroon">a</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E41:18_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">r</font>,<font color="Maroon">a</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">a3</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a3</font>,<font color="Maroon">q</font>,<font color="Maroon">r</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="ref" href="hessenbe.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>hence </b><a NAME="E2:18_3_11"/>
<font color="Maroon">a3</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E42:18_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><a NAME="E43:18_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</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">E125</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/>

<a NAME="E2:18_3_12"/><b>then </b>
( <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>, <a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E3:18_3_12"/><b>then </b><i><font color="Green">E126</font></i>: 
<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="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E4:18_3_12"/>
( <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">c3</font>,<font color="Maroon">a2</font>,<font color="Maroon">c1</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">E34</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a>, </i>;<br/>
<a NAME="E5:18_3_12"/><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> 
 
<b>by </b><i><a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E6:18_3_12"/><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> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:18_3_12"/>
contradiction
 <b>by </b><i>, <a class="txt" href="hessenbe.html#E1:18"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a>, <a class="txt" href="hessenbe.html#E6:18"><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">E127</font></i>: 
not <font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</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">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:18_3_13"/>
not <font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><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">E128</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">a</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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:18_3_14_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:18_3_14_1"/><b>then </b><i><font color="Green">E129</font></i>: 
not <font color="Maroon">x</font>,<font color="Maroon">a3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:18_3_14_1"/>
( <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:18_3_14_1"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E6:18_3_14_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">a3</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:18_3_14_1"/><b>then </b>
not <font color="Maroon">x</font>,<font color="Maroon">z</font>,<font color="Maroon">o</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">E59</font></i></a>, </i>;<br/>
<a NAME="E8:18_3_14_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:18_3_14_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E10:18_3_14_1"/><b>then </b><i><font color="Green">E130</font></i>: 
not <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:18_3_14_1"/>
<font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">a</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">E34</font></i></a>, </i>;<br/>
<a NAME="E12:18_3_14_1"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:18_3_14_1"/>
not <font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_3_14"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</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">a3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:18_3_14_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:18_3_14_2"/>
( <font color="Maroon">a3</font>,<font color="Maroon">a1</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:18_3_14_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E60</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18_3_14"/><i><font color="Green">E133</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">a3</font>,<font color="Maroon">p</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">a</font>,<font color="Maroon">o</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">a2</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">E34</font></i></a>, <a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E2:18_3_14_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">a2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:18_3_14_3"/><b>then </b><i><font color="Green">E134</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">a2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:18_3_14_3"/>
<font color="Maroon">p</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="E5:18_3_14_3"/><b>then </b>
not <font color="Maroon">p</font>,<font color="Maroon">a3</font>,<font color="Maroon">a</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">E59</font></i></a>, <a class="txt" href="hessenbe.html#E9:18"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:18_3_14_3"/>
not <font color="Maroon">a</font>,<font color="Maroon">a3</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_3_14"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:18_3_14"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</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">E137</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b2</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">E34</font></i></a>, </i>;<br/>
<a NAME="E2:18_3_14_4"/>
not <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">p</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">E58</font></i></a>, </i>;<br/>
<a NAME="E3:18_3_14_4"/><b>then </b>
not <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">p</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">E63</font></i></a>, </i>;<br/>
<a NAME="E4:18_3_14_4"/><b>then </b>
not <font color="Maroon">p</font>,<font color="Maroon">b2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:18_3_14_4"/><b>then </b>
not <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">o</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#E9:18"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:18_3_14_4"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E138</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">E139</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<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="E3:18_3_14_5"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">o</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/><a NAME="E4:18_3_14_5"/>
( <font color="Maroon">b1</font>,<font color="Maroon">c3</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">c3</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:18_3_14_5"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="ref" href="hessenbe.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E6:18_3_14_5"/><i><font color="Green">E142</font></i>: 
not <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">o</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">o</font>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <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">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</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> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E2:18_3_14_5_1"/><b>then </b>
not <font color="Maroon">b1</font>,<font color="Maroon">o</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:18_3_14_5_1"/>
not <font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E7:18_3_14_5"/>
( <font color="Maroon">b1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">z'</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">p</font>,<font color="Maroon">a3</font>,<font color="Maroon">c1</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">a3</font>,<font color="Maroon">a2</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">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:18_3_14_5_2"/><b>then </b>
<font color="Maroon">a3</font>,<font color="Maroon">c1</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:18_3_14_5_2"/>
( <font color="Maroon">b1</font>,<font color="Maroon">a3</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">z'</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">p</font>,<font color="Maroon">a3</font>,<font color="Maroon">c1</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">E34</font></i></a>, <a class="ref" href="hessenbe.html#T9" target="_self">Th9</a>, <a class="ref" href="hessenbe.html#T17" target="_self">Th17</a>, , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_3_14_5"/><b>then </b>
<font color="Maroon">c1</font>,<font color="Maroon">z'</font>,<font color="Maroon">a</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">E46</font></i></a>, , , </i>;<br/><a NAME="E9:18_3_14_5"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c1</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E10:18_3_14_5"/>
<font color="Maroon">c1</font>,<font color="Maroon">a</font>,<font color="Maroon">z</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">E34</font></i></a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E11:18_3_14_5"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c1</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E12:18_3_14_5"/>
<font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E13:18_3_14_5"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, <a class="ref" href="hessenbe.html#T8" target="_self">Th8</a>, <a class="txt" href="hessenbe.html#E6:18"><i><font color="Green">E58</font></i></a>, , , , </i>;<br/><a NAME="E14:18_3_14_5"/><i><font color="Green">E146</font></i>: 
not <font color="Maroon">a1</font>,<font color="Maroon">o</font>,<font color="Maroon">a3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E15:18_3_14_5"/>
<font color="Maroon">a1</font>,<font color="Maroon">o</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E16:18_3_14_5"/><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>, <a class="txt" href="hessenbe.html#E2:18_3"><i><font color="Green">E71</font></i></a>, </i>;<br/><a NAME="E17:18_3_14_5"/><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="E18:18_3_14_5"/><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#E1:18_3"><i><font color="Green">E70</font></i></a>, </i>;<br/><b>hence </b><a NAME="E19:18_3_14_5"/>
contradiction
 <b>by </b><i>, </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">E147</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:18_3_14_6"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E46</font></i></a>, <a class="ref" href="hessenbe.html#T19" target="_self">Th19</a>, , , <a class="txt" href="hessenbe.html#E8:18"><i><font color="Green">E60</font></i></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">E149</font></i>: 
not <font color="Maroon">q</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</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">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E2:18_3_14_6_1"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">o</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_3_14_6_1"/><b>then </b>
not <font color="Maroon">b1</font>,<font color="Maroon">q</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="hessenbe.html#T12" target="_self">Th12</a>, , </i>;<br/>
<b>hence </b><a NAME="E4:18_3_14_6_1"/>
not <font color="Maroon">q</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:18_3_14_6"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">q</font>,<font color="Maroon">b1</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">p</font>,<font color="Maroon">r</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">o</font>,<font color="Maroon">b1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E2:18_3_14_6_2"/>
( <font color="Maroon">q</font>,<font color="Maroon">b1</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b2</font>,<font color="Maroon">p</font>,<font color="Maroon">r</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="E5:18_3_14_6"/>
( <font color="Maroon">q</font>,<font color="Maroon">p</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>  &amp; <font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">r</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">z'</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">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a</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">q</font>,<font color="Maroon">a3</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">a3</font>,<font color="Maroon">r</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="ref" href="hessenbe.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>hence </b><a NAME="E2:18_3_14_6_3_1"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a</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="E2:18_3_14_6_3"/>
( <font color="Maroon">q</font>,<font color="Maroon">p</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>  &amp; <font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">b2</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b1</font>,<font color="Maroon">r</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">z'</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">E34</font></i></a>, <a class="txt" href="hessenbe.html#E3:18"><i><font color="Green">E36</font></i></a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E6:18_3_14_6"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">z'</font>,<font color="Maroon">a</font>,<font color="Maroon">c3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:18_3_14_6"/>
( <font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c3</font>,<font color="Maroon">c1</font>,<font color="Maroon">a</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">E34</font></i></a>, </i>;<br/><a NAME="E8:18_3_14_6"/><b>then </b>
<font color="Maroon">c3</font>,<font color="Maroon">z</font>,<font color="Maroon">a</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">E38</font></i></a>, </i>;<br/><a NAME="E9:18_3_14_6"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c3</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E10:18_3_14_6"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">c3</font>,<font color="Maroon">z'</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E11:18_3_14_6"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E12:18_3_14_6"/>
not <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">a</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">E58</font></i></a>, </i>;<br/><a NAME="E13:18_3_14_6"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">b1</font>,<font color="Maroon">r</font>,<font color="Maroon">z</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">E36</font></i></a>, <a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, <a class="ref" href="hessenbe.html#T16" target="_self">Th16</a>, , <a class="txt" href="hessenbe.html#E1:18_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="hessenbe.html#E2:18_3"><i><font color="Green">E71</font></i></a>, </i>;<br/><a NAME="E14:18_3_14_6"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">o</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E15:18_3_14_6"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">b2</font>,<font color="Maroon">r</font>,<font color="Maroon">p</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="E16:18_3_14_6"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b2</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">E34</font></i></a>, </i>;<br/><a NAME="E17:18_3_14_6"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">b2</font>,<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E18:18_3_14_6"/>
( <font color="Maroon">a3</font>,<font color="Maroon">a2</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">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E19:18_3_14_6"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">a3</font>,<font color="Maroon">p</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E20:18_3_14_6"/>
<font color="Maroon">b3</font>,<font color="Maroon">b2</font>,<font color="Maroon">c1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E21:18_3_14_6"/><b>then </b>
<font color="Maroon">z''</font>,<font color="Maroon">c1</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a>, <a class="txt" href="hessenbe.html#E5:18"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="hessenbe.html#T20" target="_self">Th20</a>, <a class="txt" href="hessenbe.html#E9:18"><i><font color="Green">E61</font></i></a>, <a class="txt" href="hessenbe.html#E5:18_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="hessenbe.html#E7:18_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="hessenbe.html#E9:18_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="hessenbe.html#E11:18_3"><i><font color="Green">E76</font></i></a>, </i>;<br/><a NAME="E22:18_3_14_6"/><b>then </b>
( <font color="Maroon">c1</font>,<font color="Maroon">a</font>,<font color="Maroon">z''</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c1</font>,<font color="Maroon">a</font>,<font color="Maroon">c3</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">E34</font></i></a>, </i>;<br/><a NAME="E23:18_3_14_6"/><b>then </b><i><font color="Green">E160</font></i>: 
<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="hessenbe.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E24:18_3_14_6"/><i><font color="Green">E161</font></i>: 
not <font color="Maroon">c1</font>,<font color="Maroon">c3</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E25:18_3_14_6"/>
<font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E26:18_3_14_6"/><b>then </b>
<font color="Maroon">b3</font>,<font color="Maroon">r</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="ref" href="hessenbe.html#T4" target="_self">Th4</a>, <a class="txt" href="hessenbe.html#E4:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hessenbe.html#T6" target="_self">Th6</a>, <a class="txt" href="hessenbe.html#E13:18_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="hessenbe.html#E15:18_3"><i><font color="Green">E78</font></i></a>, </i>;<br/><a NAME="E27:18_3_14_6"/><b>then </b>
( <font color="Maroon">z</font>,<font color="Maroon">r</font>,<font color="Maroon">b1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">z</font>,<font color="Maroon">r</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="hessenbe.html#E3:18_3"><i><font color="Green">E72</font></i></a>, </i>;<br/><a NAME="E28:18_3_14_6"/><b>then </b>
<font color="Maroon">z</font>,<font color="Maroon">b1</font>,<font color="Maroon">b3</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="hessenbe.html#E10:18"><i><font color="Green">E62</font></i></a>, </i>;<br/><a NAME="E29:18_3_14_6"/><b>then </b>
<font color="Maroon">b1</font>,<font color="Maroon">b3</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><a NAME="E30:18_3_14_6"/><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>, , , , , </i>;<br/><b>hence </b><a NAME="E31:18_3_14_6"/>
contradiction
 <b>by </b><i>, </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#E11:18"><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">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>
