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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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:11"/><i><font color="Green">E9</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> 
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E3:11"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</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#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E7:11"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E9:11"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E10:11"/><b>then </b><i><font color="Green">E15</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>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E11:11"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</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#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E12:11"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E13:11"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E14:11"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E15:11"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</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="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E16:11"/><b>then </b>
<font color="Maroon">c<sub>5</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#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E17:11"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:11"/>
( <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>2</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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>5</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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>4</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</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>,<font color="Maroon">c<sub>3</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<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="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>5</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>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</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>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>5</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>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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>2</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>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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#E1:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="projdes1.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="projdes1.html#E4:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="projdes1.html#E5:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="projdes1.html#E7:11"><i><font color="Green">E13</font></i></a>, <a class="txt" href="projdes1.html#E8:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="projdes1.html#E10:11"><i><font color="Green">E15</font></i></a>, <a class="txt" href="projdes1.html#E11:11"><i><font color="Green">E16</font></i></a>, <a class="txt" href="projdes1.html#E12:11"><i><font color="Green">E17</font></i></a>, <a class="txt" href="projdes1.html#E14:11"><i><font color="Green">E18</font></i></a>, <a class="txt" href="projdes1.html#E13"><i><font color="Green">Lemma6</font></i></a>, <a class="txt" href="projdes1.html#E14"><i><font color="Green">Lemma7</font></i></a></i>;<br/>


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