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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="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">X'</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y'</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">X'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X'</font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> )
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> &amp; <font color="Maroon">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">M</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">Y'</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Y'</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/>

<a NAME="E6:59"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> or  ex <font color="Olive">Xb</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">Xb</font> &amp; <font color="Olive">Xb</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) )
 
<b>by </b><i/>;<br/>
<a NAME="E7:59"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> or  ex <font color="Olive">Xc</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">Xc</font> &amp; <font color="Olive">Xc</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Xb</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E9:59"/><i><font color="Green">E37</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">Xb</font> &amp; <font color="Maroon">Xb</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">Xc</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E11:59"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Xc</font> &amp; <font color="Maroon">Xc</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E12:59"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">Xb</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Xc</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E13:59"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">Xb</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y'</font> &amp; <font color="Maroon">Xc</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y'</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E14:59"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_4.html#T42">AFF_4:42</a></i>;<br/>
<a NAME="E15:59"/>
( <font color="Maroon">Xb</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Xc</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X'</font> )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E16:59"/><b>then </b>
( <font color="Maroon">Xb</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Xc</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X'</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<a NAME="E17:59"/><b>then </b>
( <font color="Maroon">Xb</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Xc</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/>
<b>hence </b><a NAME="E18:59"/>
( <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y'</font> )
 <b>by </b><i>, , , , <a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="afproj.html#D6" target="_self">Def6</a>, </i>;<br/>


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