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<b>set </b><font color="Maroon">VV1</font> =  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>;<br/>
<b>set </b><font color="Maroon">VV2</font> =  <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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">a<sub>1</sub></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">a<sub>2</sub></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/>
<b>consider </b><font color="Maroon">P</font> being    <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><b> such that </b><br/><a NAME="E2:29_1_1"/><i><font color="Green">E25</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 ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<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/>



<b>thus </b><a NAME="E3:29_1_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> implies  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>assume </b><a NAME="E4:29_1_1"/><i><font color="Green">E26</font></i>: 
 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> )
 ;<br/>

<a NAME="E5:29_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font> &amp; <font color="Maroon">y</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="E6:29_1_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>
 <b>by </b><i>, </i>;<br/>


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