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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">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>assume </b><a NAME="E1:12"/><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; <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>  )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c</font>,<font color="Maroon">q</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i><a class="ref" href="anproj_2.html#D7">ANPROJ_2:def 7</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:12_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E3:12_2"/>
( <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:12_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E5:12_2"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">b</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, </i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:12_2_1"/><b>then </b>
( <font color="Maroon">x</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">c</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:12_2_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , <a class="ref" href="collsp.html#T11">COLLSP:11</a></i>;<br/><a NAME="E4:12_2_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:12_2_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:12_2_2"/>
( <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">x</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">c</font>,<font color="Maroon">q</font>,<font color="Maroon">y</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FCPS</font><b> such that </b><br/><a NAME="E4:12_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="anproj_2.html#D9">ANPROJ_2:def 9</a></i>;<br/><a NAME="E5:12_2_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">z</font> <a href="collsp.html#R1">is_collinear</a> 
 <b>by </b><i>, , , <a class="ref" href="anproj_2.html#D8">ANPROJ_2:def 8</a></i>;<br/><b>hence </b><a NAME="E6:12_2_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:12_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="projdes1.html#R1" target="_self">are_coplanar</a> 
 <b>by </b><i/>;<br/>


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