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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR2" target="_self">satisfies_PAP</a> 
 ;<br/>

<a NAME="E2:39"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR13" target="_self">satisfies_pap</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:39"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR1" target="_self">satisfies_PPAP</a> 
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_2.html#T24" target="_self">Th24</a></i>;<br/>
<b>thus </b><a NAME="E4:39"/>
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D4" target="_self">AFF_2:def 4</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</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>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:39_1"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:39_1"/><b>then </b><i><font color="Green">E22</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> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/>
<a NAME="E4:39_1"/><i><font color="Green">E23</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> &amp; <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="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E5:39_1"/><i><font color="Green">E24</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>9</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>9</sub></font>
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E4:39_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:39_1"/><i><font color="Green">E25</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>10</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"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:39_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E3:39_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:39_1"/><i><font color="Green">E26</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>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:39_1_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E3:39_1_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39_1"/><i><font color="Green">E27</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:39_1_4"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_4"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:39_1_4"/><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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:39_1_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>7</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="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E7:39_1_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:39_1_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E9:39_1_4"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_2.html#E3:39_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:39_1"/><i><font color="Green">E28</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_5"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:39_1_5"/><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>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_5"/><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>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E4:39_1_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E8:39_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:39_1"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_6"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:39_1_6"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:39_1"/><i><font color="Green">E30</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>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:39_1_7"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:39_1_7"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:39_1"/><i><font color="Green">E31</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>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_8"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:39_1_8"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</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="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:39_1_8"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_9"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E2:39_1_9"/><i><font color="Green">E33</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:39_1_9"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E5:39_1_9"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</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="aff_2.html#E4:39_1_9"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E6:39_1_9"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:39_1_9_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</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="aff_2.html#E4:39_1_9"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_9_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</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="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E9:39_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>16</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="E8:39_1_9"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1_9"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:39_1_9"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E11:39_1_9"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E12:39_1_9"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:39_1_9_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_9_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>18</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="E14:39_1_9"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1_9"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E15:39_1_9"/><i><font color="Green">E40</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>16</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:39_1_9_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E3:39_1_9_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:39_1_9"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E17:39_1_9"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E18:39_1_9"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>16</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="39_1_9_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:39_1_9_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_9_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:39_1_9"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>15</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="aff_2.html#E1:39_1_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>;<br/><a NAME="E20:39_1_9"/><i><font color="Green">E45</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E18:39_1_9"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/><a NAME="E21:39_1_9"/>
not <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_5"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:39_1_9_5"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E10:39_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1_9"><i><font color="Green">E42</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:39_1_9_5"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:39_1_9_5"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<a NAME="E5:39_1_9_5"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1_9_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1_9"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E19:39_1_9"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>20</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="E23:39_1_9"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/><a NAME="E24:39_1_9"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E25:39_1_9"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_6"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_6"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green">E46</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_9_6"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_9_6"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:39_1_9_6"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E10:39_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1_9"><i><font color="Green">E42</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_9_6"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E6:39_1_9_6"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:39_1_9_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_9_6"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:39_1_9"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green">E46</font></i></a>, <a class="txt" href="aff_2.html#E24:39_1_9"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E27:39_1_9"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E28:39_1_9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_7"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_7"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E2:39_1_9_7"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_9_7"/><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>16</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_7"/>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:39_1_9"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_8"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_8"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green">E46</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_9_8"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_9_8_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E2:39_1_9_8_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E3:39_1_9_8_1"/><b>then </b>
( <font color="Maroon">c<sub>20</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> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:39_1_9_8_1"/>
<font color="Maroon">c<sub>20</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="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_9_8"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E25:39_1_9"><i><font color="Green">E48</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9_8"><i><font color="Green">E52</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:39_1_9"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green">E46</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1_9"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E31:39_1_9"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E26:39_1_9"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E32:39_1_9"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E33:39_1_9"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E34:39_1_9"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E35:39_1_9"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E36:39_1_9"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1_9"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E37:39_1_9"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E30:39_1_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E38:39_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:39_1"/><b>then </b><i><font color="Green">E32</font></i>: 
not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>
<a NAME="E15:39_1"/><i><font color="Green">E33</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:39_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E18:39_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E19:39_1"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_10"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:39_1_10"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_10"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E8:39_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>16</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="E21:39_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E18:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E23:39_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E24:39_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E25:39_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_11"/>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:39_1_11"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_11"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>18</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="E27:39_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E24:39_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<a NAME="E28:39_1"/><i><font color="Green">E40</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>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_12"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:39_1_12"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E3:39_1_12"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:39_1_12"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:39_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E30:39_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</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="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E31:39_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_13"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_13"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:39_1_13"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E12:39_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:39_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>15</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="aff_2.html#E14:39_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E33:39_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_14"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<a NAME="E2:39_1_14"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E12:39_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>;<br/>
<a NAME="E34:39_1"/><i><font color="Green">E46</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E31:39_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E35:39_1"/>
not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_15"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:39_1_15"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E30:39_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="aff_2.html#E33:39_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/>
<a NAME="E3:39_1_15"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/>
<a NAME="E4:39_1_15"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<a NAME="E5:39_1_15"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_2.html#E18:39_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E32:39_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>20</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="E37:39_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/>
<a NAME="E38:39_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E39:39_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_16"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_16"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_16"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_16"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:39_1_16"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</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="aff_2.html#E30:39_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="aff_2.html#E33:39_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_16"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E6:39_1_16"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:39_1_16"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</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="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_16"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_16"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:39_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="aff_2.html#E38:39_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E41:39_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E42:39_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_17"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_17"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E2:39_1_17"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_17"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_17"/>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>10</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="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:39_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_18"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_18"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_18"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_18_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E2:39_1_18_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E3:39_1_18_1"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</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="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:39_1_18_1"/>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</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="aff_2.html#E11:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_18"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_2.html#E39:39_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:39_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="aff_2.html#E42:39_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E45:39_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E40:39_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E46:39_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E47:39_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E48:39_1"/><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>8</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E49:39_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E50:39_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E41:39_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E51:39_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E44:39_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E52:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>


</div>
