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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">K</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="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Q</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="E1:28"/><i><font color="Green">E34</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>  &amp; <font color="Maroon">Q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 ;<br/>

<a NAME="E4:28"/><i><font color="Green">E37</font></i>: 
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</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/>
<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="E6:28"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E7:28"/><i><font color="Green">E44</font></i>: 
<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">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E9:28"/><i><font color="Green">E45</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="E10:28"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </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="E12:28"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E13:28"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E14:28"/><i><font color="Green">E59</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="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:28_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/>
<a NAME="E5:28_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:28"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<a NAME="E6:28_1"/>
( <font color="Maroon">a'</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="txt" href="aff_4.html#E1:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E3:28"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E7:28_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:28"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:28_1"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="aff_4.html#E3:28"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E4:28"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:28"/>
( <font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <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_1.html#T48">AFF_1:48</a></i>;<br/>
<a NAME="E16:28"/><b>then </b>
( <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">Q</font> &amp; <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">Q</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:28"/><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">K</font>,<font color="Maroon">Q</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="E18:28"/><b>then </b>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<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">K</font>,<font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E19:28"/>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">Q</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>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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