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<div class="add">

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











<b>assume </b><a NAME="E1:69"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="diraf.html#NM2">AffinPlane</a> &amp; 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> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</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="E3:69"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="aff_4.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E5:69"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="aff_4.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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="E7:69"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E5:69"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E9:69"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:69"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E10:69"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="aff_4.html#E7:69"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E9:69"><i><font color="Green">E60</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E11:69"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:69_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E7:69"><i><font color="Green">E59</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:69_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E9:69"><i><font color="Green">E60</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E4:69_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:69"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E5:69"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E5:69_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:69"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E5:69"><i><font color="Green">E58</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>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="E13:69"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:69"><i><font color="Green">E56</font></i></a>, <a class="txt" href="aff_4.html#E10:69"><i><font color="Green">E61</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<a NAME="E14:69"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <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>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:69"><i><font color="Green">E57</font></i></a>, <a class="txt" href="aff_4.html#E5:69"><i><font color="Green">E58</font></i></a>, <a class="txt" href="aff_4.html#E7:69"><i><font color="Green">E59</font></i></a>, <a class="txt" href="aff_4.html#E9:69"><i><font color="Green">E60</font></i></a>, <a class="txt" href="aff_4.html#E13:69"><i><font color="Green">E62</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E15:69"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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