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







<b>assume </b><a NAME="E1:58"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:58"/>
for <font color="Olive">A</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><br/> for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>  st <font color="Olive">A</font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Olive">C</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/><font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<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>let </b><font color="Maroon">C</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#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4" target="_self">//</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <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#T19" target="_self">Th19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><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="E1:58_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green">E52</font></i></a>, , , <a class="ref" href="aff_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:58_1_1_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">x'</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:58_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">x'</font> <a href="aff_1.html#R2" target="_self">//</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green">E52</font></i></a>, </i>;<br/><b>consider </b><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="E6:58_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E7:58_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2" target="_self">Line</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="aff_1.html#E30"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="aff_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:58_1_1_1"/>
<font color="Maroon">p</font>,<font color="Maroon">x'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green">E52</font></i></a>, , , <a class="txt" href="aff_1.html#E30"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="aff_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E9:58_1_1_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">x'</font>
 <b>by </b><i/>;<br/><a NAME="E10:58_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1" target="_self">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:58_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T25" target="_self">Th25</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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