<?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="E1:41"/><i><font color="Green">E19</font></i>: 
 <a href="diraf.html#R1">Mid</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="E2:41"/><i><font color="Green">E20</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:41"/><i><font color="Green">E21</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>6</sub></font>
 ;<br/>

<a NAME="E4:41"/><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>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</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>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</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>5</sub></font> )
 ;<br/><b>then 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="E3:41_1"/><i><font color="Green">E25</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:41_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pasch.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E5:41_1"/><i><font color="Green">E27</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E3:41_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E6:41_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>7</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#T34">DIRAF:34</a></i>;<br/><a NAME="E7:41_1"/><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>3</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E5:41_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="diraf.html#T18">DIRAF:18</a></i>;<br/><a NAME="E8:41_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E3:41_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#T15">DIRAF:15</a></i>;<br/><a NAME="E9:41_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E10:41_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E5:41_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="diraf.html#T16">DIRAF:16</a></i>;<br/><a NAME="E11:41_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</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="E13:41_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:41_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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>by </b><i><a class="txt" href="pasch.html#E7:41_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E15:41_1"/><i><font color="Green">E32</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E13:41_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E16:41_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</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="E17:41_1"/><i><font color="Green">E34</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>4</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><a NAME="E18:41_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:41_1_1"/><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>3</sub></font> &amp;  <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>6</sub></font> &amp;  <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>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <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="E3:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E17:41_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:41_1"/><i><font color="Green">E36</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="41_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_2"/>
<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="E2:41_1_2"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E13:41_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:41_1_2"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:41_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</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="E22:41_1"/><i><font color="Green">E37</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E23:41_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pasch.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E24:41_1"/><i><font color="Green">E39</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_3"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:41_1_3"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:41_1"/><i><font color="Green">E40</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_4"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:41_1_4"/><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>5</sub></font> &amp;  <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>3</sub></font> &amp;  <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>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <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="E3:41_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E17:41_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pasch.html#E24:41_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:41_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E27:41_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</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="txt" href="pasch.html#E4:41"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pasch.html#E13:41_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E28:41_1"/>
<font color="Maroon">c<sub>5</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>8</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:41_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pasch.html#E14:41_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pasch.html#E18:41_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E29:41_1"/><b>then </b>
<font color="Maroon">c<sub>5</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>9</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E19:41_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pasch.html#E23:41_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E30:41_1"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font> or  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="diraf.html#T11">DIRAF:11</a></i>;<br/><div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_5"/><i><font color="Green">E45</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</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="E3:41_1_5"/><i><font color="Green">E46</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:41_1_5"/><i><font color="Green">E47</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E5:41_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pasch.html#E25:41_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pasch.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:41_1_5"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:41_1_5_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:41_1_5"><i><font color="Green">E45</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E3:41_1_5_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:41_1_5_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T41">DIRAF:41</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:41_1_5"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E3:41_1_5"><i><font color="Green">E46</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/><a NAME="E7:41_1_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</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#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E8:41_1_5"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><a NAME="E9:41_1_5"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E15:41_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pasch.html#E22:41_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/><a NAME="E10:41_1_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</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/><a NAME="E11:41_1_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E27:41_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pasch.html#E4:41_1_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><a NAME="E12:41_1_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</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/><a NAME="E13:41_1_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E5:41_1_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><a NAME="E14:41_1_5"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E3:41_1_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pasch.html#E8:41_1_5"><i><font color="Green">E49</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><b>hence </b><a NAME="E15:41_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T18">DIRAF:18</a></i>;<br/></div><b>end;</b></div><a NAME="E32:41_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E30:41_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E33:41_1"/><i><font color="Green">E46</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> or  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E15:41_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pasch.html#E22:41_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="diraf.html#T21">DIRAF:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_6"/><i><font color="Green">E47</font></i>: 
not  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:41_1_6"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</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="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/><a NAME="E3:41_1_6"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E22:41_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pasch.html#E33:41_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pasch.html#E1:41_1_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="diraf.html#T15">DIRAF:15</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</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="E5:41_1_6"/><i><font color="Green">E49</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</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>10</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:41_1_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pasch.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E6:41_1_6"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_6_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:41_1_6_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E22:41_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E3:41_1_6_1"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E16:41_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:41_1_6_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</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="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T41">DIRAF:41</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E27:41_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pasch.html#E3:41_1_6_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:41_1_6"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E32:41_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E8:41_1_6"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E9:41_1_6"/><b>then </b>
<font color="Maroon">c<sub>9</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>4</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E10:41_1_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:41_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pasch.html#E26:41_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E11:41_1_6"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E5:41_1_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pasch.html#E6:41_1_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E12:41_1_6"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E14:41_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pasch.html#E18:41_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E13:41_1_6"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E14:41_1_6"/><b>then </b>
 <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>10</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/><a NAME="E15:41_1_6"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E16:41_1_6"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</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="41_1_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_6_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:41_1_6_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</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>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:41_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pasch.html#E14:41_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E3:41_1_6_2"/><b>then </b>
( <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>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E18:41_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E4:41_1_6_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</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="pasch.html#E4:41"><i><font color="Green">E22</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E5:41_1_6_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E6:41_1_6_2"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11">DIRAF:11</a></i>;<br/>
<a NAME="E7:41_1_6_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E8:41_1_6_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</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/>
<a NAME="E9:41_1_6_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:41"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:41_1_6"/>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E6:41_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pasch.html#E5:41_1_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E18:41_1_6"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_6_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:41_1_6_3"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E5:41_1_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><b>hence </b><a NAME="E3:41_1_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T18">DIRAF:18</a></i>;<br/></div><b>end;</b></div><a NAME="E20:41_1_6"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <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>3</sub></font> &amp;  <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>8</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</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="pasch.html#E16:41_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pasch.html#E15:41_1_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pasch.html#E18:41_1_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/><a NAME="E21:41_1_6"/><b>then </b>
(  <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> &amp;  <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>2</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E16:41_1_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T41">DIRAF:41</a></i>;<br/><b>hence </b><a NAME="E22:41_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E3:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E19:41_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="pasch.html#E20:41_1_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/></div><b>end;</b></div><a NAME="E35:41_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><b>hence </b><a NAME="E36:41_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E30:41_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pasch.html#E31:41_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:41"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>
<a NAME="E7:41"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>
<a NAME="E8:41"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:41"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E5:41"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pasch.html#E6:41"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E7:41"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
