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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green">E25</font></i>: 
 <a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font> is <a href="incproj.html#V6">2-dimensional</a>
 ;<br/>

<b>assume </b><a NAME="E2:69"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">AS</font> is  not   <a href="diraf.html#NM2">AffinPlane</a>
 ;<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>;<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="E5:69"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E6:69"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37">AFF_4:37</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E8:69"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="aff_4.html#T48">AFF_4:48</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>;<br/>
<a NAME="E9:69"/><i><font color="Green">E34</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</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>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="afproj.html#D6" target="_self">Def6</a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <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">X</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">K</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> as   <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> <b>by </b><i><a class="ref" href="afproj.html#D5" target="_self">Def5</a>, </i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <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><b> such that </b><br/><a NAME="E12:69"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/>
<a NAME="E13:69"/>
<font color="Maroon">a</font> is  not   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>
 
<b>by </b><i><a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="afproj.html#T10" target="_self">Th10</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E15:69"/><i><font color="Green">E37</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">Y'</font> &amp; <font color="Maroon">Y'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E16:69"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y'</font> <a href="aff_4.html#R1">'||'</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="afproj.html#T9" target="_self">Th9</a>, <a class="ref" href="afproj.html#T10" target="_self">Th10</a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a>, , </i>;<br/>
<a NAME="E17:69"/><b>then </b>
<font color="Maroon">Y'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="afproj.html#T9" target="_self">Th9</a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/>
<a NAME="E18:69"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="afproj.html#T12" target="_self">Th12</a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/>
<a NAME="E19:69"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="afproj.html#D5" target="_self">Def5</a>, <a class="ref" href="afproj.html#T9" target="_self">Th9</a>, <a class="ref" href="aff_4.html#T43">AFF_4:43</a></i>;<br/>
<b>hence </b><a NAME="E20:69"/>
contradiction
 <b>by </b><i><a class="ref" href="afproj.html#D6" target="_self">Def6</a>, <a class="ref" href="afproj.html#T9" target="_self">Th9</a></i>;<br/>


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