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

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











<b>assume </b><a NAME="E1:5"/><i><font color="Green">E23</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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">a</font>,<font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:5_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:5"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:5_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:5_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:5"/><i><font color="Green">E26</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_3"/><i><font color="Green">E28</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:5_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:5_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:5_3"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E5:5_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:5_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:5_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:5"/><i><font color="Green">E29</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:5_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:5_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:5"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E8:5"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E9:5"/><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>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E11:5"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E12:5"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E13:5"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E14:5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E15:5"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E16:5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:5_5"/><b>then </b>
<font color="Maroon">q</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="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:5_5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:5_5"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">q</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:5_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_6"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:5_6"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_6_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:5_6_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E3:5_6_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E4:5_6_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:5_6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/></div><b>end;</b></div><a NAME="E5:5_6"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_6_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:5_6_2"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:5_6_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:5_6_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:5_6"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E7:5_6"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E8:5_6"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E9:5_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E10:5_6"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E11:5_6"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:5"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Olive">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> )
 <b>by </b><i>, </i>;<br/>


</div>
