<?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="E1:25"/><i><font color="Green">E8</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:25"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E5:25"/><i><font color="Green">E12</font></i>: 
not  <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#E1:25"><i><font color="Green">E8</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>
<a NAME="E6:25"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:25"><i><font color="Green">E9</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E7:25"/><b>then </b><i><font color="Green">E13</font></i>: 
 <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>5</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E8:25"/>
<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>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:25"><i><font color="Green">E10</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E4:25"><i><font color="Green">E11</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:25"/><i><font color="Green">E16</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="diraf.html#T17">DIRAF:17</a></i>;<br/>
<a NAME="E13:25"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E12:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E14:25"/><i><font color="Green">E18</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>2</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:25"><i><font color="Green">E8</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:25"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E17:25"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E13:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E18:25"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E16:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:25_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>7</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T34">DIRAF:34</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E12:25"><i><font color="Green">E16</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="E5:25_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="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="E6:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:25"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pasch.html#E12:25"><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="E19:25"/><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>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_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="E2:25_2"/><b>then </b>
<font color="Maroon">c<sub>4</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#E17:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E3:25_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</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>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E13:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E4:25_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E12:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E5:25_2"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:25_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:25"><i><font color="Green">E8</font></i></a>, <a class="ref" href="diraf.html#T36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:25"/>
<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>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E13:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<a NAME="E21:25"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:25"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pasch.html#E14:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</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="E23:25"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E24:25"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E12:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E25:25"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="diraf.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#E16:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E26:25"/><b>then </b>
( <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>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E23:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="diraf.html#T27">DIRAF:27</a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E27:25"/><b>then </b>
( <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>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E28:25"/><b>then </b>
<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>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="pasch.html#E19:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E29:25"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E30:25"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="diraf.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#E17:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E24:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E31:25"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.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#E18:25"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T28">DIRAF:28</a></i>;<br/>
<a NAME="E32:25"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E5:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pasch.html#E7:25"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pasch.html#E9:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pasch.html#E10:25"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pasch.html#E17:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pasch.html#E24:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pasch.html#E29:25"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pasch.html#T11" target="_self">Th11</a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/>
<b>hence </b><a NAME="E33:25"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pasch.html#E18:25"><i><font color="Green">E21</font></i></a>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>


</div>
