<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> )
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>assume </b><a NAME="E3:80"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:80"/><i><font color="Green" title="E68">A3</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E7:80"/><i><font color="Green" title="E69">A4</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <b>and </b><br/><a NAME="E8:80"/><i><font color="Green" title="E70">A5</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:80"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="aff_4.html#E47"><i><font color="Green" title="E40">Lm9</font></i></a></i>;<br/>
<a NAME="E9:80"/><i><font color="Green" title="E71">A6</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E8:80"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>;<br/>
<b>set </b><font color="Maroon" title="c8">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>;<br/>
<a NAME="E10:80"/><i><font color="Green" title="E72">A7</font></i>: 
( <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<a NAME="E11:80"/><i><font color="Green" title="E73">A8</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="aff_4.html#E9:80"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E12:80"/><i><font color="Green" title="E74">A9</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:80"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:80"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="aff_4.html#E7:80"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="aff_4.html#E9:80"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<a NAME="E13:80"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:80"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:80"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="aff_4.html#E11:80"><i><font color="Green" title="E73">A8</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a></i>;<br/>
<a NAME="E14:80"/><b>then </b><i><font color="Green" title="E75">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E10:80"><i><font color="Green" title="E72">A7</font></i></a>, <a class="txt" href="aff_4.html#E11:80"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="aff_4.html#E42"><i><font color="Green" title="E35">Lm8</font></i></a></i>;<br/>
<a NAME="E15:80"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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="c4">a</font>,<font color="Maroon" title="c6">c</font>
 ;<br/>

<a NAME="E2:80_1"/><b>then </b>
( <font color="Maroon" title="c4">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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">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="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</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="c4">a</font>,<font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E8:80"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="aff_4.html#E11:80"><i><font color="Green" title="E73">A8</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:80"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:80"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:80"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="aff_4.html#E11:80"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="aff_4.html#E12:80"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="aff_4.html#E14:80"><i><font color="Green" title="E75">A10</font></i></a>, <a class="ref" href="aff_4.html#T26" target="_self" title="AFF_4:th.26">Th26</a></i>;<br/>


</div>
