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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">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<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>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:52"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <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="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:52"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:52"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:52"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="ref" href="aff_4.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E10:52"/>
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, , </i>;<br/>
<a NAME="E11:52"/><b>then </b>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a>, , <a class="ref" href="aff_4.html#T10" target="_self">Th10</a>, <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:52"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i>, <a class="ref" href="aff_4.html#T10" target="_self">Th10</a></i>;<br/>


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