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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">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> or <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E5:31"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<b>consider </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>, <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="E7:31"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E8:31"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E9:31"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:31_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E3:31_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="txt" href="aff_4.html#E2:31"><i><font color="Green">E35</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font>
 ;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:31_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:31_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:31"><i><font color="Green">E37</font></i></a>, </i>;<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="E6:31_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 <b>and </b><br/><a NAME="E7:31_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<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="E9:31_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E10:31_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E11:31_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:31_2_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:31_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:31_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:31_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:31_2_1"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/><a NAME="E13:31_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="txt" href="aff_4.html#E5:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><a NAME="E14:31_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/><a NAME="E15:31_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:31_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="txt" href="aff_4.html#E5:31"><i><font color="Green">E43</font></i></a>, <a class="txt" href="aff_4.html#E7:31"><i><font color="Green">E44</font></i></a>, </i>;<br/><a NAME="E17:31_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a></i>;<br/><b>hence </b><a NAME="E18:31_2_1"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , <a class="txt" href="aff_4.html#E3:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E8:31"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:31_2"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:31"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:31"><i><font color="Green">E34</font></i></a></i>;<br/>


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