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<b>let </b><font color="Maroon" title="c1">FCPS</font> be  <a href="anproj_2.html#NV10" title="ANPROJ_2:NV.10">up-3-dimensional</a> <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">d'</font>, <font color="Maroon" title="c6">s</font>, <font color="Maroon" title="c7">a'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>;<br/>













<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E3">A1</font></i>: 
( not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">d'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c6">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d'</font> &amp; <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c5">d'</font>,<font color="Maroon" title="c6">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">a'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a'</font> )
 ;<br/>

<b>assume </b><a NAME="E2:5"/>
not <font color="Maroon" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font>
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">d'</font>,<font color="Maroon" title="c7">a'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/>
<a NAME="E4:5"/>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">d'</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/>
<a NAME="E5:5"/><b>then </b>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">a'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="projdes1.html#E3:5"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></i>;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c4">d</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></i>;<br/>
<a NAME="E7:5"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c3">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="projdes1.html#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">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" title="E3">A1</font></i></a>, <a class="txt" href="projdes1.html#E6:5"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="collsp.html#T11" title="COLLSP:th.11">COLLSP:11</a></i>;<br/>


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