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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><b>that </b><br/><a NAME="E1:1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">AS</font> is   <a href="diraf.html#NM2">AffinPlane</a>
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</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>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <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="E4:1"/><i><font color="Green">E30</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T21">AFF_1:21</a></i>;<br/>
<a NAME="E5:1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">K</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E6:1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E7:1"/><i><font color="Green">E34</font></i>: 
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">Y</font> =  <a href="aff_4.html#K1">Plane</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E8:1"/><i><font color="Green">E35</font></i>: 
 <a href="aff_4.html#K1">Plane</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 
<b>by </b><i>, , <a class="ref" href="aff_4.html#D2">AFF_4:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><b>set </b><font color="Maroon">K'</font> = <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>;<br/><a NAME="E3:1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="aff_4.html#T27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3">AFF_4:def 3</a></i>;<br/><a NAME="E4:1_2"/><b>then </b>
not <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>then 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="E6:1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="aff_4.html#K2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E7:1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E8:1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">zz</font> where <font color="Olive">zz</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> :  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">zz</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1">Plane</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="aff_4.html#D1">AFF_4:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1">Plane</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:1"/>
<font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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