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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="E1:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">AS</font> is   <a href="diraf.html#NM2">AffinPlane</a>
 ;<br/>

<b>assume </b><a NAME="E2:23"/>
not  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font> <a href="xboole_0.html#R1">misses</a>  <a href="afproj.html#K8" target="_self">Dir_of_Planes</a> <font color="Maroon">AS</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font> &amp; <font color="Maroon">x</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><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:23"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X'</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font> ;<br/>
<a NAME="E7:23"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X'</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 
<b>by </b><i>, </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="E9:23"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E10:23"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i>, </i>;<br/>
<b>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="E12:23"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E13:23"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E15:23"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E16:23"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<b>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="E18:23"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font>
 <b>and </b><br/><a NAME="E19:23"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y'</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>and </b><br/><a NAME="E20:23"/>
<font color="Maroon">X</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E21:23"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X'</font>
 
<b>by </b><i>, <a class="txt" href="afproj.html#E1:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_4.html#T33">AFF_4:33</a></i>;<br/>
<a NAME="E22:23"/>
not <font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y'</font>
 
;<br/>
<b>hence </b><a NAME="E23:23"/>
contradiction
 <b>by </b><i><a class="ref" href="afproj.html#T1" target="_self">Th1</a>, <a class="txt" href="afproj.html#E4:23"><i><font color="Green">E26</font></i></a></i>;<br/>


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