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<b>let </b><font color="Maroon">P</font> be    <a href="relset_1.html#M2">Relation</a> of  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>, <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>, <font color="Maroon">Q</font> be    <a href="relset_1.html#M2">Relation</a> of  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>, <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff  ex <font color="Olive">A</font>, <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">A</font> &amp; <font color="Olive">y</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">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">A</font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">X</font> ) )
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff  ex <font color="Olive">A</font>, <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">A</font> &amp; <font color="Olive">y</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">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">A</font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">X</font> ) )
 ;<br/>

<a NAME="E3:29_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:29_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff  ex <font color="Olive">A</font>, <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#K5" target="_self">LDir</a> <font color="Olive">A</font> &amp; <font color="Maroon">y</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">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">A</font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">X</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:29_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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