<?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>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E4:29"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E5:29"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E6:29"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</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="homothet.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="txt" href="homothet.html#E2:29"><i><font color="Green">E31</font></i></a>, <a class="txt" href="homothet.html#E3:29"><i><font color="Green">E32</font></i></a>, <a class="txt" href="homothet.html#E4:29"><i><font color="Green">E33</font></i></a>, <a class="txt" href="homothet.html#E29"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E7:29"/><i><font color="Green">E36</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>
<a NAME="E8:29"/><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>7</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E4:29"><i><font color="Green">E33</font></i></a>, <a class="txt" href="homothet.html#E6:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
 <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>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E37</font></i>: 
 <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>9</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</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="E12:29"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E7:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E13:29"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E14:29"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:29"><i><font color="Green">E32</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/>
<a NAME="E15:29"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E16:29"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E4:29"><i><font color="Green">E33</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E18:29"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</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#E5:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E15:29"><i><font color="Green">E40</font></i></a>, <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E19:29"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E10:29"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:29"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="txt" href="homothet.html#E4:29"><i><font color="Green">E33</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div>
