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

<b>let </b><font color="Maroon">c<sub>1</sub></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">E87</font></i>: 
 <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="collsp.html#V4">proper</a>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><a NAME="E3:97"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></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">b<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a>  );<br/>
<a NAME="E5:97"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:97_1_1"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E2:97_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:97"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span> st not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">b<sub>6</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  ) ) holds <br/>ex <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>5</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> 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">c<sub>1</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_2"/><i><font color="Green">E93</font></i>: 
( not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E2:97_2"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E3:97_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:97_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:97_2"/>
(  not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_2_1"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<a NAME="E2:97_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E3:97_2_1"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E4:97_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:97_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_2"><i><font color="Green">E94</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:97"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> 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">c<sub>1</sub></font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:97_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>and </b><br/><a NAME="E2:97_3"/><i><font color="Green">E95</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_3"/><i><font color="Green">E97</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E5:97_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E6:97_3"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:97_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:97_3"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>;<br/><div><i><font color="Green">E101</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">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:97_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">E102</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:97_3_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E4:97_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></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">E104</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">E105</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E2:97_3_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_1_1"><i><font color="Green">E105</font></i></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">E105</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E2:97_3_1_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_1_2"><i><font color="Green">E105</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:97_3_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/><a NAME="E7:97_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></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">E105</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">E106</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><a NAME="E2:97_3_1_2_2_1"/>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_2_1"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_2_1"><i><font color="Green">E106</font></i></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">E106</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><a NAME="E2:97_3_1_2_2_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:97_3_1_2_2_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2_2_2"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_2"><i><font color="Green">E105</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>16</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:97_3_1_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E11:97_3_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3"><i><font color="Green">E94</font></i></a></i>;<br/><a NAME="E12:97_3_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><div><i><font color="Green">E108</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">E109</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:97_3_1_2_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E3:97_3_1_2_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E4:97_3_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:97_3_1_2_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_3_1_2_3"><i><font color="Green">E110</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2_3"><i><font color="Green">E111</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E6:97_3_1_2_3"/>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:97_3_1_2_3"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E5:97_3_1_2_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E9:97_3_1_2_3"/>
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E10:97_3_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E11:97_3_1_2_3"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="anproj_2.html#E8:97_3_1_2_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E12:97_3_1_2_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:97_3_1_2_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E13:97_3_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2_3"><i><font color="Green">E114</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">E109</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:97_3_1_2_4"/>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <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">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_3_1_2_4_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E3:97_3_1_2_4_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:97_3_1_2_4_1_1"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:97_3_1_2_4_1"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:97_3_1_2_4_1_2"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</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#E2:97_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_3_1_2_4"><i><font color="Green">E109</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_3_1_2_4_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2_4_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:97_3_1_2_4"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="anproj_2.html#E12:97_3_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="anproj_2.html#E92"><i><font color="Green">Lemma85</font></i></a></i>;<br/><a NAME="E5:97_3_1_2_4"/>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:97_3_1_2_4"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3_1_2_4"><i><font color="Green">E110</font></i></a>, <a class="ref" href="anproj_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E8:97_3_1_2_4"/>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:97_3_1_2_4"><i><font color="Green">E110</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3_1_2_4"><i><font color="Green">E111</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E10:97_3_1_2_4"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E11:97_3_1_2_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>20</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:97_3_1_2_4_2"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:97_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3_1_2_4"><i><font color="Green">E111</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:97_3_1_2_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E2:97_3_1_2_4_3"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E5:97_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1_2_4_3"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_3_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="anproj_2.html#E10:97_3_1_2_4"><i><font color="Green">E112</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:97_3_1_2_4"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E10:97_3_1_2_4"><i><font color="Green">E112</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E14:97_3_1_2_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:97_3_1_2_4"><i><font color="Green">E113</font></i></a>, <a class="txt" href="anproj_2.html#E12:97_3_1_2_4"><i><font color="Green">E114</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:97_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">E108</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:97_3_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_3_1"><i><font color="Green">E101</font></i></a></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:97"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">b<sub>6</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  ) ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> 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">c<sub>1</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>and </b><br/><a NAME="E2:97_4"/><i><font color="Green">E96</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_4"/><i><font color="Green">E97</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E4:97_4"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:97_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97"><i><font color="Green">E92</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_4"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_4"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E7:97_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E8:97_4"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:97"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_4"><i><font color="Green">E95</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_4"><i><font color="Green">E99</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:97_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E7:97"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_4"><i><font color="Green">E97</font></i></a>, <a class="txt" href="anproj_2.html#E6:97_4"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_4"><i><font color="Green">E100</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:97"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:97_5"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>and </b><br/><a NAME="E2:97_5"/><i><font color="Green">E97</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<a NAME="E3:97_5"/><i><font color="Green">E98</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E7:97"><i><font color="Green">E93</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_5"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_5"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E4:97_5"/>
( not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; ( for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>(  not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  or  not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  ) ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:97"><i><font color="Green">E94</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_5"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_5"><i><font color="Green">E97</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:97_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E5:97"><i><font color="Green">E91</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_5"><i><font color="Green">E98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:97"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; not <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> 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">c<sub>1</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>and </b><br/><a NAME="E2:97_6"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:97_6"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></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">c<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E2:97_6_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_6"><i><font color="Green">E97</font></i></a></i>;<br/><div><i><font color="Green">E101</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">E102</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:97_6_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E3:97_6_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6"><i><font color="Green">E98</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E1:97_6_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E4:97_6_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green">E100</font></i></a></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">E102</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:97_6_1_2"/>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:97_6_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E5:97_6_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6"><i><font color="Green">E98</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E6:97_6_1_2"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E7:97_6_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="anproj_2.html#E5:97_6_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E8:97_6_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97_6_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_6_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E9:97_6_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97_6_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="anproj_2.html#E4:97_6_1_2"><i><font color="Green">E103</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:97_6_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_6_1"><i><font color="Green">E101</font></i></a></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:97"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] ) holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] &amp; not <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:97_7"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:97_7"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:97_7"/><i><font color="Green">E100</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E87</font></i></a>, <a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green">E99</font></i></a>, <a class="ref" href="collsp.html#T19">COLLSP:19</a></i>;<br/>
<a NAME="E6:97_7"/><i><font color="Green">E101</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E5:97_7"><i><font color="Green">E100</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E7:97_7"/><i><font color="Green">E102</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_7_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E2:97_7_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green">E99</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E3:97_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green">E99</font></i></a>, <a class="txt" href="anproj_2.html#E5:97_7"><i><font color="Green">E100</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:97_7"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:97_7"><i><font color="Green">E99</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E9:97_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:97_7"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E10:97_7"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E10:97"><i><font color="Green">E96</font></i></a>, <a class="txt" href="anproj_2.html#E7:97_7"><i><font color="Green">E102</font></i></a>, <a class="txt" href="anproj_2.html#E8:97_7"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:97_7"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:97_7"><i><font color="Green">E101</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:97"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span>;<br/>





<a NAME="E1:97_8"/>
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E3:97"><i><font color="Green">E89</font></i></a>, <a class="ref" href="anproj_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E2:97_8"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="anproj_2.html#E4:97"><i><font color="Green">E90</font></i></a>, <a class="txt" href="anproj_2.html#E9:97"><i><font color="Green">E95</font></i></a></i>;<br/>
<a NAME="E3:97_8"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E11:97"><i><font color="Green">E97</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:97_8"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="anproj_2.html#E9:97"><i><font color="Green">E95</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:97"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> 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">c<sub>1</sub></font></span>)</span>;<br/>









<a NAME="E1:97_9"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E12:97"><i><font color="Green">E98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:97_9"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="anproj_2.html#NM1">CollProjectiveSpace</a> <b>by </b><i><a class="txt" href="anproj_2.html#E1:97"><i><font color="Green">E87</font></i></a>, <a class="txt" href="anproj_2.html#E2:97"><i><font color="Green">E88</font></i></a>, <a class="ref" href="anproj_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E15:97"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="anproj_1.html#K6">ProjectiveSpace</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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#E13:97"><i><font color="Green">E99</font></i></a>, <a class="ref" href="anproj_2.html#D15" target="_self">Def15</a></i>;<br/>


</div>
