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

<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E3:80"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E5:80"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:80"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E8:80"/><i><font color="Green">E43</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:80"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E10:80"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E11:80"/><i><font color="Green">E47</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E12:80"/><i><font color="Green">E49</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, , , </i>;<br/>
<a NAME="E13:80"/><b>then </b>
( <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="txt" href="aff_4.html#E25"><i><font color="Green">Lemma68</font></i></a>, , <a class="txt" href="aff_4.html#E7:80"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E14:80"/><b>then </b><i><font color="Green">E57</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="txt" href="aff_4.html#E2:80"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E15:80"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">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">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:80_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">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">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T33">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#E24"><i><font color="Green">Lemma67</font></i></a>, , , , </i>;<br/>


</div>
