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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E15</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green">E17</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E19</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:21"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T17">DIRAF:17</a></i>;<br/>
<a NAME="E8:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E9:21"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E12:21"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E8:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E13:21"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E15:21"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:21"><i><font color="Green">E23</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E16:21"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E17:21"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E18:21"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E12:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E3:21_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E7:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="papdesaf.html#E14:21"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:21"><i><font color="Green">E15</font></i></a>, <a class="txt" href="papdesaf.html#E7:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="papdesaf.html#E14:21"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:21"/><i><font color="Green">E28</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E29</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:21_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E3:21_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E7:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="papdesaf.html#E14:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="papdesaf.html#E1:21_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<a NAME="E4:21_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:21"><i><font color="Green">E15</font></i></a>, <a class="txt" href="papdesaf.html#E17:21"><i><font color="Green">E26</font></i></a>, <a class="txt" href="papdesaf.html#E18:21"><i><font color="Green">E27</font></i></a>, <a class="txt" href="papdesaf.html#E3:21_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:21"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E21:21"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<a NAME="E1:21_3"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E2:21_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<b>hence </b><a NAME="E3:21_3"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="papdesaf.html#E9:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:21"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<a NAME="E1:21_4"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E12:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E2:21_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:21_4"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="papdesaf.html#E9:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:21"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<a NAME="E1:21_5"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:21"><i><font color="Green">E23</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E2:21_5"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E3:21_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E4:21_5"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E3:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="papdesaf.html#E9:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:21"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E19:21"><i><font color="Green">E28</font></i></a>, <a class="txt" href="papdesaf.html#E21:21"><i><font color="Green">E30</font></i></a>, <a class="txt" href="papdesaf.html#E22:21"><i><font color="Green">E31</font></i></a>, <a class="ref" href="papdesaf.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E25:21"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:21"><i><font color="Green">E23</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<b>hence </b><a NAME="E26:21"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E18:21"><i><font color="Green">E27</font></i></a>, <a class="txt" href="papdesaf.html#E24:21"><i><font color="Green">E32</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/>

<a NAME="E27:21"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E12:21"><i><font color="Green">E24</font></i></a>, <a class="txt" href="papdesaf.html#E24:21"><i><font color="Green">E32</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<b>hence </b><a NAME="E28:21"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E20:21"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/>


</div>
