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<div class="add">

<b>let </b><font color="Maroon">V</font> be  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green">E58</font></i>: 
 <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font> is <a href="collsp.html#V4">proper</a>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">r</font> &amp; <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">r</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D10" target="_self">ANPROJ_2:def 10</a><br/>

<b>given </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><a NAME="E3:97"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">q3</font>, <font color="Olive">r3</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">r1</font>,<font color="Olive">r2</font>,<font color="Olive">r3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Olive">r3</font>,<font color="Olive">q3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>] <b>means </b>for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">x2</font>, <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">y1</font>,<font color="Olive">y2</font>,<font color="Olive">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">x1</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  );<br/>
<a NAME="E5:97"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><a NAME="E1:97_1_1"/>
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">y2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E7"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E2:97_1_1"/>
 ex <font color="Olive">x2</font>, <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:97"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  holds <br/>(  not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  ) ) holds <br/> ex <font color="Olive">q3</font>, <font color="Olive">p3</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">q3</font>,<font color="Olive">p3</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_2"/><i><font color="Green">E70</font></i>: 
( not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E2:97_2"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  holds <br/>(  not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E3:97_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">q1</font>,<font color="Maroon">q2</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">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:97_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:97_2"/>
(  not <font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">q1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_2_1"/>
( <font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:97_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E3:97_2_1"/>
<font color="Maroon">q</font>,<font color="Maroon">q1</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#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:97_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:97_2"/>
 ex <font color="Olive">q3</font>, <font color="Olive">p3</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">p3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">q3</font>,<font color="Olive">p3</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:97"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>] &amp; not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:97_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 <b>and </b><br/><a NAME="E2:97_3"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_3"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E5:97_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E6:97_3"/><i><font color="Green">E80</font></i>: 
( <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">q</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:97_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 ;<br/><a NAME="E2:97_3_1_1"/><b>then </b>
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">q</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_1"/>
 ex <font color="Olive">z2</font>, <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font>
 ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E3:97_3_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">b'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">b</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a></i>;<br/><a NAME="E4:97_3_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 ;<br/><b>take </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font>;<br/><a NAME="E2:97_3_1_2_1_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="anproj_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><b>take </b><font color="Maroon">p2</font> = <font color="Maroon">p2</font>;<br/><a NAME="E2:97_3_1_2_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="anproj_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i>, <a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E6:97_3_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> )
 ;<br/><a NAME="E7:97_3_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y1</font>
 ;<br/><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>;<br/><a NAME="E2:97_3_1_2_2_1"/>
<font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_2_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="anproj_2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font>
 ;<br/><b>take </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font>;<br/><a NAME="E2:97_3_1_2_2_2"/>
<font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_2_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="anproj_2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E12:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">x3</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E9:97_3_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">x3</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">d'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E11:97_3_1_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">x1</font>,<font color="Maroon">x3</font>,<font color="Maroon">d'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">d</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">d'</font>,<font color="Maroon">d</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a></i>;<br/><a NAME="E12:97_3_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, , <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_3"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:97_3_1_2_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">b'</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><a NAME="E3:97_3_1_2_3"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d'</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_2.html#T6" target="_self">Th6</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><a NAME="E4:97_3_1_2_3"/>
<font color="Maroon">b</font>,<font color="Maroon">q</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#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:97_3_1_2_3"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">b'</font>,<font color="Maroon">d'</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#T7" target="_self">Th7</a>, <a class="txt" href="anproj_2.html#E19"><i><font color="Green">Lemma111</font></i></a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:97_3_1_2_3"/>
<font color="Maroon">d'</font>,<font color="Maroon">x3</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E8:97_3_1_2_3"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E20"><i><font color="Green">Lemma112</font></i></a>, <a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E9:97_3_1_2_3"/>
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E10:97_3_1_2_3"/><b>then </b>
( <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">y1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E12:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self">Def1</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E11:97_3_1_2_3"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D1" target="_self">Def1</a>, <a class="txt" href="anproj_2.html#E21"><i><font color="Green">Lemma113</font></i></a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E12:97_3_1_2_3"/>
<font color="Maroon">q</font>,<font color="Maroon">z1</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E21"><i><font color="Green">Lemma113</font></i></a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>hence </b><a NAME="E13:97_3_1_2_3"/>
 ex <font color="Olive">z2</font>, <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_2.html#E22"><i><font color="Green">Lemma114</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_3_1_2_4"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:97_3_1_2_4"/>
not <font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3_1_2_4_1"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_3_1_2_4_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E3:97_3_1_2_4_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E2:97_3_1_2_4_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:97_3_1_2_4_1"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_1_2"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E2:97_3_1_2_4_1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:97_3_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_2.html#T6" target="_self">Th6</a>, <a class="ref" href="anproj_2.html#T7" target="_self">Th7</a>, <a class="txt" href="anproj_2.html#E19"><i><font color="Green">Lemma111</font></i></a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">o</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E4:97_3_1_2_4"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">b'</font>,<font color="Maroon">d'</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">o</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_2.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E5:97_3_1_2_4"/>
<font color="Maroon">d'</font>,<font color="Maroon">x3</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E7:97_3_1_2_4"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T7" target="_self">Th7</a>, <a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E8:97_3_1_2_4"/>
( <font color="Maroon">x1</font>,<font color="Maroon">o</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">o</font>,<font color="Maroon">x</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#T7" target="_self">Th7</a>, <a class="txt" href="anproj_2.html#E19"><i><font color="Green">Lemma111</font></i></a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E10:97_3_1_2_4"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">x1</font>,<font color="Maroon">x</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">z1</font>,<font color="Maroon">q</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E11:97_3_1_2_4"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_2"/>
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:97_3_1_2_4_2"/><b>then </b>
( <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">y1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b'</font>,<font color="Maroon">x3</font>,<font color="Maroon">y2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E12:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E13:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self">Def1</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_2"/>
<font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D1" target="_self">Def1</a>, <a class="txt" href="anproj_2.html#E19"><i><font color="Green">Lemma111</font></i></a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:97_3_1_2_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:97_3_1_2_4_3"/>
( <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:97_3_1_2_4_3"/><b>then </b>
( <font color="Maroon">x1</font>,<font color="Maroon">x</font>,<font color="Maroon">p1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x1</font>,<font color="Maroon">x</font>,<font color="Maroon">p2</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_3"/>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_2.html#E20"><i><font color="Green">Lemma112</font></i></a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:97_3_1_2_4"/>
<font color="Maroon">q</font>,<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E20"><i><font color="Green">Lemma112</font></i></a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>hence </b><a NAME="E14:97_3_1_2_4"/>
 ex <font color="Olive">z2</font>, <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E21"><i><font color="Green">Lemma113</font></i></a>, <a class="txt" href="anproj_2.html#E22"><i><font color="Green">Lemma114</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:97_3_1_2"/>
 ex <font color="Olive">z2</font>, <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E16"><i><font color="Green">Lemma108</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:97_3_1"/>
 ex <font color="Olive">z2</font>, <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:97_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:97"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>] &amp; not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  holds <br/>(  not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  ) ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_4"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 <b>and </b><br/><a NAME="E2:97_4"/><i><font color="Green">E171</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_4"/><i><font color="Green">E172</font></i>: 
not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_4"/><i><font color="Green">E173</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  holds <br/>(  not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">q3</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">p3</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E6:97_4"/><i><font color="Green">E174</font></i>: 
( <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; not <font color="Maroon">q3</font>,<font color="Maroon">p3</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E7"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, , </i>;<br/>
<a NAME="E7:97_4"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">q3</font>,<font color="Maroon">p3</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Maroon">p3</font>,<font color="Maroon">p3</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:97_4"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">q3</font>,<font color="Maroon">p3</font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E9:97_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:97"/><i><font color="Green">E176</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>] &amp; not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/>for <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:97_5"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 <b>and </b><br/><a NAME="E2:97_5"/><i><font color="Green">E178</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>



<a NAME="E3:97_5"/><i><font color="Green">E311</font></i>: 
( not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E4:97_5"/>
( not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  holds <br/>(  not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  ) ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E9"><i><font color="Green">Lemma94</font></i></a>, <a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E5:97_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:97"/><i><font color="Green">E312</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">x</font>, <font color="Olive">q3</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>] &amp; not <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">q</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">q</font>,<font color="Olive">q3</font>,<font color="Olive">x</font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q3</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q3</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_6"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 <b>and </b><br/><a NAME="E2:97_6"/><i><font color="Green">E314</font></i>: 
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_6"/><i><font color="Green">E315</font></i>: 
( <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">q3</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E2:97_6_1"/><i><font color="Green">E316</font></i>: 
( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/><div><i><font color="Green">E317</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_6_1_1"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 ;<br/><a NAME="E2:97_6_1_1"/><b>then </b>
( <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">q3</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">z1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><a NAME="E3:97_6_1_1"/><b>then </b>
<font color="Maroon">q3</font>,<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E4:97_6_1_1"/>
 ex <font color="Olive">x2</font>, <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_6_1_2"/><i><font color="Green">E319</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 ;<br/><a NAME="E2:97_6_1_2"/>
( <font color="Maroon">q3</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E4:97_6_1_2"/><i><font color="Green">E320</font></i>: 
( <font color="Maroon">q3</font>,<font color="Maroon">z1</font>,<font color="Maroon">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">z2</font>,<font color="Maroon">x2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E5:97_6_1_2"/><i><font color="Green">E321</font></i>: 
<font color="Maroon">q1</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#T1" target="_self">Th1</a></i>;<br/><a NAME="E6:97_6_1_2"/>
( <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:97_6_1_2"/><b>then </b>
( <font color="Maroon">x</font>,<font color="Maroon">z2</font>,<font color="Maroon">q1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">z2</font>,<font color="Maroon">q2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , <a class="ref" href="anproj_2.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E8:97_6_1_2"/><b>then </b>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">x2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E9:97_6_1_2"/>
 ex <font color="Olive">x2</font>, <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Olive">x1</font>,<font color="Olive">x2</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="E5:97_6_1"/>
 ex <font color="Olive">x2</font>, <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Olive">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q3</font>,<font color="Olive">x1</font>,<font color="Olive">x2</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:97_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q3</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:97"/><i><font color="Green">E322</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  st ( for <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>] ) holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>] &amp; not <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">p</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:97_7"/><i><font color="Green">E323</font></i>: 
for <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Olive">q1</font>,<font color="Olive">q2</font>]
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E3:97_7"/><i><font color="Green">E324</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E5:97_7"/><i><font color="Green">E325</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="ref" href="collsp.html#T19">COLLSP:19</a></i>;<br/>
<a NAME="E6:97_7"/><i><font color="Green">E326</font></i>: 
not <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E7:97_7"/><i><font color="Green">E327</font></i>: 
not <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_7_1"/>
<font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_7_1"/><b>then </b>
( <font color="Maroon">q</font>,<font color="Maroon">x1</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">x1</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#T1" target="_self">Th1</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:97_7_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:97_7"/><i><font color="Green">E328</font></i>: 
( <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">x1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i>, <a class="ref" href="anproj_2.html#T1" target="_self">Th1</a>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E9:97_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E10:97_7"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">x1</font>,<font color="Maroon">x2</font>]
 
<b>by </b><i><a class="ref" href="anproj_2.html#T4" target="_self">Th4</a>, , </i>;<br/>
<b>hence </b><a NAME="E11:97_7"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>] &amp; not <font color="Olive">p1</font>,<font color="Olive">p2</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="E12:97"/><i><font color="Green">E329</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>





<a NAME="E1:97_8"/>
not <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E2"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E2:97_8"/><b>then </b>
for <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Olive">p1</font>,<font color="Olive">p2</font>]
 
<b>by </b><i>, <a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a></i>;<br/>
<a NAME="E3:97_8"/><b>then </b>
 ex <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">r1</font>,<font color="Olive">r2</font>] &amp; not <font color="Olive">r1</font>,<font color="Olive">r2</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:97_8"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>,<font color="Maroon">z</font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E10"><i><font color="Green">Lemma95</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:97"/><i><font color="Green">E330</font></i>: 
for <font color="Olive">p4</font>, <font color="Olive">p1</font>, <font color="Olive">q</font>, <font color="Olive">q4</font>, <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>  ex <font color="Olive">r</font>, <font color="Olive">r1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Olive">p4</font>,<font color="Olive">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">p1</font>,<font color="Olive">q4</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">r2</font>,<font color="Olive">r</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p4</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">q4</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>









<a NAME="E1:97_9"/>
 ex <font color="Olive">r1</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p4</font>,<font color="Maroon">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q4</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r2</font>,<font color="Olive">r</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:97_9"/>
 ex <font color="Olive">r</font>, <font color="Olive">r1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( <font color="Maroon">p4</font>,<font color="Maroon">q</font>,<font color="Olive">r</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">p1</font>,<font color="Maroon">q4</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">r2</font>,<font color="Olive">r</font>,<font color="Olive">r1</font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">CS</font> =  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font> as   <a href="anproj_2.html#NM1">CollProjectiveSpace</a> <b>by </b><i>, , <a class="txt" href="anproj_2.html#E3:97"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">CS</font>
;<br/>

<b>thus </b><a NAME="E15:97"/>
( <font color="Maroon">CS</font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">V</font> &amp; <font color="Maroon">CS</font> is <a href="anproj_2.html#V8" target="_self">at_most-3-dimensional</a> )
 <b>by </b><i>, <a class="txt" href="anproj_2.html#E7:97"><i><font color="Green">E74</font></i></a></i>;<br/>


</div>
