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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<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>assume </b><a NAME="E1:83"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:83"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E4:83"/><i><font color="Green">E36</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:83"/><i><font color="Green">E37</font></i>: 
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</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">p</font>,<font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/>
<a NAME="E6:83"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E7:83"/><i><font color="Green">E44</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</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/>
<b>set </b><font color="Maroon">M'</font> = <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">p</font>,<font color="Maroon">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">N'</font> = <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">p</font>,<font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E8:83"/><i><font color="Green">E45</font></i>: 
( <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">p</font>,<font color="Maroon">q</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">p</font>,<font color="Maroon">r</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">p</font>,<font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">p</font>,<font color="Maroon">r</font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#R4">//</a> <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">p</font>,<font color="Maroon">q</font></span>)</span> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="aff_1.html#R4">//</a> <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">p</font>,<font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E10:83"/><i><font color="Green">E47</font></i>: 
( <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">p</font>,<font color="Maroon">q</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">p</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E11:83"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E6:83"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E12:83"/><i><font color="Green">E57</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</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">p</font>,<font color="Maroon">r</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="E13:83"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:83_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:83_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>
<a NAME="E14:83"/><b>then </b>
<font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</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>, , , , , </i>;<br/>
<b>hence </b><a NAME="E15:83"/>
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Maroon">X</font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i>, , </i>;<br/>


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