<?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><a NAME="E1:39"/><i><font color="Green">E18</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> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; 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="E2:39"/><b>then </b><i><font color="Green">E19</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>6</sub></font>,<font color="Maroon">c<sub>5</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/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E21</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>3</sub></font> )
 ;<br/><a NAME="E2:39_1"/>
<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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:39"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</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="E4:39_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</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>7</sub></font> &amp; <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>7</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green">E23</font></i>: 
 <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>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:39_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E6:39_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>5</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E7:39_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</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#E1:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pasch.html#E1:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T16">DIRAF:16</a></i>;<br/><a NAME="E8:39_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <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>7</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E9:39_1"/><i><font color="Green">E26</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="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_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="E2:39_1_1"/><b>then </b>
 <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>5</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pasch.html#E7:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:39_1"/><i><font color="Green">E27</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>7</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pasch.html#E5:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><a NAME="E11:39_1"/><i><font color="Green">E28</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="39_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2"/>
 <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="E2:39_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 <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>2</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E3:39_1_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>7</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E8:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E4:39_1_2"/><b>then </b>
(  <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>5</sub></font> &amp;  <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>7</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E5:39_1"><i><font color="Green">E23</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="E5:39_1_2"/><b>then </b>
(  <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>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>7</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>5</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>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<a NAME="E6:39_1_2"/><b>then </b>
(  <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;  <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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E10:39_1"><i><font color="Green">E27</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="E7:39_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pasch.html#E9:39_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pasch.html#E2:39_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><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="E13:39_1"/><i><font color="Green">E29</font></i>: 
(  <a href="diraf.html#R1">Mid</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; <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>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E8:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pasch.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E14:39_1"/>
<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="txt" href="pasch.html#E11:39_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><a NAME="E15:39_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>8</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:39_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pasch.html#E13:39_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E16:39_1"/><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>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E17:39_1"/><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>8</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E18:39_1"/><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>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>8</sub></font>,<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#E13:39_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><b>hence </b><a NAME="E19:39_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>2</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="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E22</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#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E2:39_2"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<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#E1:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pasch.html#E1:39_2"><i><font color="Green">E22</font></i></a>, <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="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E2:39_3"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <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>7</sub></font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pasch.html#E1:39_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/><b>thus </b><a NAME="E3:39_3"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>7</sub></font>,<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#E1:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pasch.html#E1:39_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T13">DIRAF:13</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
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>2</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="Olive">b<sub>1</sub></font>,<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#E3:39"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E4:39"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
