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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="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:49"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">X</font> =  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font>;<br/><a NAME="E4:49_1"/>
not <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="aff_4.html#E6:49_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E5:49_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E6:49_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E7:49_1"/>
<font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E8:49_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a>, </i>;<br/><b>hence </b><a NAME="E9:49_1"/>
 ex <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</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, , <a class="ref" href="aff_4.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>consider </b><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="E3:49_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E5:49_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">X</font> =  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font>;<br/><a NAME="E6:49_2"/>
not <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, <a class="txt" href="aff_4.html#E1:49_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E7:49_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="aff_4.html#E1:49_2"><i><font color="Green">E47</font></i></a>, </i>;<br/><a NAME="E8:49_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">P</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49_2"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:49_2"/>
 ex <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</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i>, <a class="ref" href="aff_4.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:49"/>
 ex <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</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i/>;<br/>


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