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<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>let </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>, <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><b>that </b><br/><a NAME="E1:70"/><i><font color="Green">E66</font></i>: 
<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="E2:70"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:70_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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#E1:70_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:70_1"/>
<font color="Maroon">c<sub>7</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="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_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>2</sub></font>,<font color="Maroon">c<sub>4</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="Maroon">c<sub>6</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#E2:70_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:70_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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="70_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <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>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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#E1:70"><i><font color="Green">E66</font></i></a>, <a class="txt" href="anproj_2.html#E2:70"><i><font color="Green">E67</font></i></a>, <a class="txt" href="anproj_2.html#E1:70_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:70_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:70_2_1"/>
<font color="Maroon">c<sub>7</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="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_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>2</sub></font>,<font color="Maroon">c<sub>4</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="Maroon">c<sub>6</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#E2:70_2_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E2:70_2_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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="70_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_2_2_1"/>
( <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>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<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="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#E1:70"><i><font color="Green">E66</font></i></a>, <a class="txt" href="anproj_2.html#E3:70"><i><font color="Green">E68</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="E2:70_2_2_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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#E1:70_2_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="anproj_2.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:70_2_2"/>
<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> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:70_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>2</sub></font>,<font color="Maroon">c<sub>4</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="Maroon">c<sub>6</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#E2:70_2_2"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:70_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>2</sub></font>,<font color="Maroon">c<sub>4</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="Maroon">c<sub>6</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#E1:70_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="anproj_2.html#E2:70_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:70"/>
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>2</sub></font>,<font color="Maroon">c<sub>4</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="Maroon">c<sub>6</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#E4:70"><i><font color="Green">E69</font></i></a></i>;<br/>


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