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



<b>assume </b><a NAME="E1:46"/>
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 ;<br/>

<b>then 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="E3:46"/><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; not <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E35</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/>;<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="E6:46"/><i><font color="Green">E36</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> )
 <b>and </b><br/><a NAME="E7:46"/><i><font color="Green">E37</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_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font>;<br/>
<a NAME="E8:46"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E10:46"/><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">c</font>
 <b>and </b><br/><a NAME="E11:46"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<a NAME="E12:46"/><i><font color="Green">E47</font></i>: 
<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">P</font>
 
<b>by </b><i><a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E13:46"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><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></i>;<br/>

<a NAME="E14:46"/>
<font color="Maroon">a</font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">K</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><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="txt" href="aff_4.html#E10:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="aff_4.html#E12:46"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E15:46"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_4.html#T8" target="_self">Th8</a></i>;<br/>

<b>thus </b><a NAME="E16:46"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:46_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</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="txt" href="aff_4.html#E8:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:46_1"/>
contradiction
 <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="txt" href="aff_4.html#E10:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="aff_4.html#E11:46"><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#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>


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