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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>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E1:27"/><i><font color="Green">E23</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>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; 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>,<font color="Maroon">c<sub>6</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</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>  &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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>3</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>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c<sub>9</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>4</sub></font>,<font color="Maroon">c<sub>6</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</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> <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>9</sub></font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b>
not <font color="Maroon">c<sub>6</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="ref" href="projdes1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:27"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<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="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E24</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>6</sub></font>
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E6:27"/><b>then </b><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E4:27"><i><font color="Green">E24</font></i></a>, <a class="ref" href="projdes1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  )
 
<b>by </b><i><a class="ref" href="projdes1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E8:27"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E9:27"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E7:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projdes1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="projdes1.html#E7:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projdes1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>6</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>by </b><i><a class="txt" href="projdes1.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="projdes1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E12:27"/>
not <font color="Maroon">c<sub>3</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> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E13:27"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</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="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:27"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E15:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</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="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E16:27"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</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="projdes1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E17:27"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="projdes1.html#E11:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projdes1.html#E14:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projdes1.html#E15:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projdes1.html#E9"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E18:27"/><b>then </b>
not <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="collsp.html#R1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E6:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projdes1.html#E9:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projdes1.html#E10:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="projdes1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E19:27"/>
not <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</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/>


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