<?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="E1:18"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_3.html#NR2" target="_self">satisfies_DES1_1</a> 
 ;<br/>

<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_3.html#D1" target="_self">AFF_3:def 1</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>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E2:18"/><i><font color="Green">E10</font></i>: 
( <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>  )
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E12</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>6</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>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>8</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> &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>10</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>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>8</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>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</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>7</sub></font> )
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E13</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>6</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>11</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>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E6:18"/>
( <font color="Maroon">c<sub>8</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>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</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>10</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E14</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</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>8</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>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</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>8</sub></font>,<font color="Maroon">c<sub>6</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>8</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>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E15</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>9</sub></font> &amp; <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> &amp; <font color="Maroon">c<sub>7</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="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E16</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E10:18"/><i><font color="Green">E17</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E11:18"/><i><font color="Green">E18</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E12:18"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</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>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E13:18"/><i><font color="Green">E20</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E14:18"/><i><font color="Green">E21</font></i>: 
not  <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>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
 <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>10</sub></font>
 ;<br/>

<a NAME="E2:18_1"/><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_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:18"/><i><font color="Green">E22</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>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:18_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</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>7</sub></font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="aff_3.html#E14:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_3"/><i><font color="Green">E24</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:18_3"/>
(  <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>9</sub></font>,<font color="Maroon">c<sub>12</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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E3:18_3"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18"><i><font color="Green">E9</font></i></a>, <a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E8:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="aff_3.html#E11:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="aff_3.html#E12:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="aff_3.html#E13:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_3.html#E14:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_3.html#E15:18"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_3.html#E1:18_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="aff_3.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4"/><i><font color="Green">E24</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</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="E2:18_4"/><i><font color="Green">E25</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E3:18_4"/><i><font color="Green">E26</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</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>11</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>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>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</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>7</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> &amp; <font color="Maroon">c<sub>13</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_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="aff_3.html#E8:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E4:18_4"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:18_4_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</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>7</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E1:18_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E3:18_4_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:18_4_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</sub></font>,<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_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_3.html#E2:18_4_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E5:18_4_1"/><b>then </b>
<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>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><a NAME="E6:18_4_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E10:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E7:18_4_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:18_4_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>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E3:18_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</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_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:18_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_4_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_3.html#E6:18_4_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E9:18_4_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_4_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:18_4_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</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_3.html#E2:18_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E3:18_4_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E4:18_4_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>8</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_3.html#E5:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E4:18_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="aff_3.html#E2:18_4_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E5:18_4_2"/><b>then </b>
<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>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><a NAME="E6:18_4_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E9:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E7:18_4_2"/>
 <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4_2_1"/>
 <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="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:18_4_2_1"/><b>then </b>
 <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>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E3:18_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</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_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E15:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:18_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="aff_3.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_4_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E7:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_3.html#E3:18_4_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_3.html#E6:18_4_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E9:18_4_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:18_4"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E4:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="aff_3.html#E5:18_4"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:18"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="aff_3.html#E16:18"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
