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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">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span>;<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="E2:49"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><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="E4:49"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">X'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E6:49"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>and </b><br/><a NAME="E7:49"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T5" target="_self">Th5</a>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<a NAME="E8:49"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">x</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E10:49"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">X'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">x</font> <a href="aff_4.html#K2">*</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T5" target="_self">Th5</a>, <a class="ref" href="afproj.html#T6" target="_self">Th6</a>, <a class="ref" href="aff_4.html#T38">AFF_4:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">Z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">AS</font></span>)</span> <b>by </b><i>, <a class="ref" href="afproj.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<a NAME="E12:49"/>
( <font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">X'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="afproj.html#T5" target="_self">Th5</a>, , , <a class="ref" href="aff_4.html#T41">AFF_4:41</a>, <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/>
<b>hence </b><a NAME="E13:49"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="afproj.html#T5" target="_self">Th5</a>, , </i>;<br/>


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