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<b>let </b><font color="Maroon">AP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AP</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:70"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="txt" href="aff_1.html#E3:70"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E4:70"/><i><font color="Green">E50</font></i>: 
 <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:70"/>
not  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1"/>
 <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green">E52</font></i>: 
(  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E15"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E4:70_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:70_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T25" target="_self">Th25</a>, <a class="txt" href="aff_1.html#E16"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:70_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T25" target="_self">Th25</a>, <a class="txt" href="aff_1.html#E2:70"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font><b> such that </b><br/><a NAME="E7:70"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="txt" href="aff_1.html#E30"><i><font color="Green">Lemma57</font></i></a>, </i>;<br/>
<a NAME="E8:70"/>
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T25" target="_self">Th25</a>, </i>;<br/>
<b>hence </b><a NAME="E9:70"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AP</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T25" target="_self">Th25</a></i>;<br/>


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