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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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E5:25_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</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="E7:25_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:25_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">X</font> = <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E9:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E10:25_1"/>
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>given </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><a NAME="E1:25_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:25_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 ;<br/><a NAME="E3:25_2"/>
 <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:25_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">2</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> iff  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">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Olive">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  ) )
 <b>by </b><i/>;<br/>


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