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<b>let </b><font color="Maroon">FCPS</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font>;<br/>



















<b>assume </b><a NAME="E1:23"/><i><font color="Green">E25</font></i>: 
( not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="collsp.html#R1">is_collinear</a>  &amp; not <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">p</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">r</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a'</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:23"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="collsp.html#R1">is_collinear</a> 
 ;<br/>

<a NAME="E3:23"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">c</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">b</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">c</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:23"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font>,<font color="Maroon">a'</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:23"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">a'</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:23"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


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