<?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="E1:20"/><i><font color="Green">E7</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="E2:20"/><i><font color="Green">E8</font></i>: 
 <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>and </b><br/><a NAME="E3:20"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E10</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>
 ;<br/>

<div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:20_1"/><b>then </b>
(  <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>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>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="ref" href="diraf.html#T24">DIRAF:24</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E3:20_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:20"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pasch.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E4:20_1"/>
 <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>by </b><i><a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E12</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>
 ;<br/><a NAME="E2:20_2"/><b>then </b><i><font color="Green">E13</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#E1:20"><i><font color="Green">E7</font></i></a>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><a NAME="E3:20_2"/><i><font color="Green">E14</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>5</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E4:20_2"/><i><font color="Green">E15</font></i>: 
not  <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>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_1"/>
 <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>5</sub></font>
 ;<br/>

<a NAME="E2:20_2_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E3:20_2_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</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#E3:20_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<a NAME="E4:20_2_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <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>5</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E5:20_2_1"/>
 <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>by </b><i><a class="txt" href="pasch.html#E1:20"><i><font color="Green">E7</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E6:20_2_1"/><b>then </b>
(  <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="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="E7:20_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pasch.html#E1:20_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pasch.html#E4:20_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:20_2"/><b>then </b><i><font color="Green">E16</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>5</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E6:20_2"/>
 <a href="diraf.html#R3">LIN</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#E2:20"><i><font color="Green">E8</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E7:20_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E8:20_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<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="E10:20_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E11:20_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="diraf.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="pasch.html#E5:20_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="diraf.html#T32">DIRAF:32</a></i>;<br/><a NAME="E12:20_2"/><i><font color="Green">E19</font></i>: 
not  <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>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2"/><i><font color="Green">E20</font></i>: 
 <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>6</sub></font>
 ;<br/>

<a NAME="E2:20_2_2"/><b>then </b><i><font color="Green">E21</font></i>: 
(  <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>6</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="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E3:20_2_2"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:20_2_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:20_2_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="diraf.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#D5">DIRAF:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:20_2_2_1"/><b>then </b>
<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>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><a NAME="E3:20_2_2_1"/><b>then </b>
 <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>5</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><b>hence </b><a NAME="E4:20_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:20_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pasch.html#E4:20_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pasch.html#E2:20_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:20_2_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>4</sub></font> &amp;  <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>by </b><i><a class="txt" href="pasch.html#E1:20"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pasch.html#E2:20"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pasch.html#E4:20_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E7:20_2_2"/><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>3</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>4</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>5</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="E8:20_2_2"/><b>then </b>
 <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>5</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:20_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E9:20_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:20_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pasch.html#E4:20_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pasch.html#E2:20_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:20_2"/><i><font color="Green">E20</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>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_2_3"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.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#E10:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E2:20_2_3"/><b>then </b>
 <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>7</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E3:20_2_3"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:20_2_3"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E11:20_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E5:20_2_3"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pasch.html#E3:20_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E6:20_2_3"/>
 <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#E1:20"><i><font color="Green">E7</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>
<b>hence </b><a NAME="E7:20_2_3"/>
 <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>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E12:20_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pasch.html#E3:20_2_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E5:20_2_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pasch.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E14:20_2"/><i><font color="Green">E21</font></i>: 
not  <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>
 ;<br/><a NAME="E15:20_2"/><i><font color="Green">E22</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>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:20_2_4"/><b>then </b>
(  <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> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:20"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E3:20_2_4"/><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>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <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#T36">DIRAF:36</a></i>;<br/>
<a NAME="E4:20_2_4"/><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>2</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>5</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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:20_2"><i><font color="Green">E13</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="E5:20_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pasch.html#E2:20_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:20_2"/><i><font color="Green">E23</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_5"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:20_2_5"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:20"><i><font color="Green">E8</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/>
<a NAME="E3:20_2_5"/><b>then </b><i><font color="Green">E24</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E15:20_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>
<a NAME="E4:20_2_5"/><i><font color="Green">E25</font></i>: 
<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>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E11:20_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:20_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E10:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><a NAME="E3:20_2_5_1"/><b>then </b>
 <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>2</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E4:20_2_5_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</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>6</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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:20"><i><font color="Green">E8</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="E5:20_2_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pasch.html#E15:20_2"><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="E6:20_2_5"/><b>then </b>
( <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> &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>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>5</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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E3:20_2_5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E4:20_2_5"><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="E7:20_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_6"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:20_2_6"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.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#E10:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/><a NAME="E3:20_2_6"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</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>7</sub></font> )
 <b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/><a NAME="E4:20_2_6"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <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>5</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E11:20_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/><a NAME="E5:20_2_6"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E16:20_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pasch.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:20_2_6"/><b>then </b>
 <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>7</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E7:20_2_6"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E13:20_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T18">DIRAF:18</a></i>;<br/><b>hence </b><a NAME="E8:20_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/></div><b>end;</b></div><a NAME="E18:20_2"/><b>then </b>
(  <a href="diraf.html#R1">Mid</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>2</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#E1:20"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pasch.html#E2:20"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pasch.html#E14:20_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a>, <a class="ref" href="diraf.html#T35">DIRAF:35</a></i>;<br/><a NAME="E19:20_2"/><b>then </b>
 <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>4</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E3:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pasch.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E20:20_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:20"><i><font color="Green">E7</font></i></a>, <a class="ref" href="diraf.html#T18">DIRAF:18</a></i>;<br/><b>hence </b><a NAME="E21:20_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E4:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:20"/>
 <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>by </b><i><a class="txt" href="pasch.html#E5:20"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
