<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">SAS</font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>





<b>assume </b><a NAME="E1:49"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E3:49"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E30</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:49"/>
( not <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="txt" href="semi_af1.html#E9:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E8:49"/><i><font color="Green">E31</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<a NAME="E9:49"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="txt" href="semi_af1.html#E3:49"><i><font color="Green">E29</font></i></a>, <a class="txt" href="semi_af1.html#E5:49"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E10:49"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 
<b>by </b><i>, <a class="txt" href="semi_af1.html#E8:49"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E11:49"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="semi_af1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:49"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; not <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E13:49"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:49"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
