<?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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E6</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>4</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>3</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>2</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>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</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>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E4:5"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 ;<br/>

<a NAME="E5:5"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>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> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E7:5"/><i><font color="Green">E11</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</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>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="translac.html#T2">TRANSLAC:2</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>6</sub></font>,<font color="Maroon">c<sub>4</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>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E8:5"/><i><font color="Green">E12</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<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>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>4</sub></font> &amp; <font color="Maroon">c<sub>4</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>4</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>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E7:5"><i><font color="Green">E11</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="E9:5"/><i><font color="Green">E13</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<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>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E5:5"><i><font color="Green">E10</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="E10:5"/><i><font color="Green">E14</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<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#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E5:5"><i><font color="Green">E10</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/>
<b>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="E12:5"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</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>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E13:5"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E14:5"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/>
<a NAME="E3:5_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</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>4</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/>
<a NAME="E4:5_1"/><b>then </b>
<font color="Maroon">c<sub>2</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E5:5_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</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="E16:5"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E13:5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T73">AFF_1:73</a></i>;<br/>
<a NAME="E17:5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</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="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="txt" href="homothet.html#E16:5"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E18:5"/><i><font color="Green">E19</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:5_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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E3:5_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:5"/><i><font color="Green">E20</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:5_3"/><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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E3:5_3"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E7:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:5"/><i><font color="Green">E21</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:5_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E3:5_4"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E4:5_4"/><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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E5:5_4"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E7:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<a NAME="E6:5_4"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E7:5_4"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:5"/><i><font color="Green">E22</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_5"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:5_5"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:5_5"/><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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E2:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:5_5"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E7:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:5_5"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:5"/><i><font color="Green">E23</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_6"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:5_6"/><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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E3:5_6"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E7:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:5_6"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="homothet.html#E21:5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:5"/><i><font color="Green">E24</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E24:5"/><i><font color="Green">E25</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<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="5_7"><b>proof </b></a><div class="add">

<a NAME="E1:5_7"/>
( <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>2</sub></font>,<font color="Maroon">c<sub>4</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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:5_7"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:5"/><i><font color="Green">E26</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<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="5_8"><b>proof </b></a><div class="add">

<a NAME="E1:5_8"/>
( <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>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:5_8"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</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>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_9"><b>proof </b></a><div class="add">

<a NAME="E1:5_9"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E2:5_9"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/>
<a NAME="E3:5_9"/><b>then </b>
<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>15</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E3:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/>
<a NAME="E4:5_9"/><b>then </b>
<font color="Maroon">c<sub>8</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="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="txt" href="homothet.html#E13:5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<b>hence </b><a NAME="E5:5_9"/>
<font color="Maroon">c<sub>4</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>16</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="homothet.html#E12:5"><i><font color="Green">E15</font></i></a>, <a class="txt" href="homothet.html#E16:5"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:5"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_10"><b>proof </b></a><div class="add">

<a NAME="E1:5_10"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E2:5_10"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E4:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="homothet.html#E16:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="homothet.html#E22:5"><i><font color="Green">E23</font></i></a>, <a class="txt" href="homothet.html#E23:5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="homothet.html#E26:5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="homothet.html#E3:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="homothet.html#E4:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="homothet.html#E16:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="homothet.html#E18:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="homothet.html#E19:5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="homothet.html#E24:5"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E29:5"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="homothet.html#E16:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="homothet.html#E17:5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="homothet.html#E20:5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="homothet.html#E21:5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="homothet.html#E24:5"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E25:5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>


</div>
