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<b>let </b><font color="Maroon">CLSP</font> be  <a href="collsp.html#V4" target="_self">proper</a> <a href="collsp.html#NM2">CollSp</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="collsp.html#M2" target="_self">LINE</a> of <font color="Maroon">CLSP</font>;<br/>









<b>assume </b><a NAME="E1:33"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font>, <font color="Maroon">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E4:33"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:33"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> : <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">d</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green">E67</font></i>: 
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">x</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  )
 
;<br/>
<a NAME="E7:33"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> : <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">d</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E68</font></i>: 
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">y</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  )
 
;<br/>
<a NAME="E9:33"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> : <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">d</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:33"/><b>then </b>
 ex <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CLSP</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">z</font> <a href="collsp.html#R1" target="_self">is_collinear</a>  )
 
;<br/>
<b>hence </b><a NAME="E11:33"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="collsp.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="ref" href="collsp.html#D2" target="_self">Def2</a>, <a class="txt" href="collsp.html#E5"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="collsp.html#E6"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>


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