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<b>let </b><font color="Maroon" title="c1">AP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E55">A2</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<a NAME="E3:68"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:68"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="aff_1.html#E2:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="aff_1.html#T47" target="_self" title="AFF_1:th.47">Th47</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<a NAME="E4:68"/><i><font color="Green" title="E57">A4</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="txt" href="aff_1.html#E3:68"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:68"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E58">A5</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D5" target="_self" title="AFF_1:def.5">Def5</a></i>;<br/>
<a NAME="E4:68_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" target="_self" title="AFF_1:th.26">Th26</a></i>;<br/>
<a NAME="E5:68_1"/><b>then </b>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E3:68_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="aff_1.html#T36" target="_self" title="AFF_1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E6:68_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_1.html#E2:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="aff_1.html#E3:68_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="aff_1.html#T46" target="_self" title="AFF_1:th.46">Th46</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E7:68"/><i><font color="Green" title="E58">A6</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E1:68"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="aff_1.html#E4:68"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="aff_1.html#T72" target="_self" title="AFF_1:th.72">Th72</a></i>;<br/>
<a NAME="E8:68"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="aff_1.html#E7:68"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:68"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E7:68"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/>


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