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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="anproj_2.html#V7">up-3-dimensional</a> <a href="anproj_2.html#NM1">CollProjectiveSpace</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>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><a NAME="E1:5"/><i><font color="Green">E3</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>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>6</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>4</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>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:5"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</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="projdes1.html#E1:5"><i><font color="Green">E3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:5"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="projdes1.html#E1:5"><i><font color="Green">E3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:5"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</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="projdes1.html#E1:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="projdes1.html#E3:5"><i><font color="Green">E4</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:5"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green">E3</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E8:5"/>
contradiction
 <b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="projdes1.html#E6:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/>


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