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<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> 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="E1:18"/><i><font color="Green">E18</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>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E19</font></i>: 
 <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>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E4:18"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E22</font></i>: 
<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>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:18"/><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>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E7:18"/>
<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>
 <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"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<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>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="translac.html#E4:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<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#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="translac.html#E6:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


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

<a NAME="E8:18"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="translac.html#E3:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="translac.html#E5:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E9:18"/>
 <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>
 <b>by </b><i><a class="txt" href="translac.html#E1:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="translac.html#E2:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>


</div>
