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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="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E26</font></i>: 
<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>
 ;<br/><b>reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E3:21_1"/>
<font color="Maroon">x''</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">AS</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">AS</font>
 <b>and </b><br/><a NAME="E6:21_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x''</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</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="E8:21_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:21_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/>;<br/><b>take </b><font color="Maroon">X</font> = <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E10:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , </i>;<br/><b>thus </b><a NAME="E11:21_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="21_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="E1:21_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</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="E2:21_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="E3:21_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:21_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">AS</font>
 <b>by </b><i/>;<br/><a NAME="E5:21_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">AS</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:21_2"/>
<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>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <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> 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>  <a href="afproj.html#K6" target="_self">PDir</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/>;<br/>


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