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<b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">OAS</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR6" target="_self">is_Dil</a> 
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:80"/><i><font color="Green">E26</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E30</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E31</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T43">DIRAF:43</a></i>;<br/><a NAME="E4:80_1"/><i><font color="Green">E34</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">r</font>,<font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , <a class="ref" href="transgeo.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E5:80_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:80_1_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">r</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">q</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#T36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="transgeo.html#T35" target="_self">Th35</a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:80_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="transgeo.html#D4" target="_self">Def4</a>, <a class="ref" href="transgeo.html#T30" target="_self">Th30</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:80"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="transgeo.html#T30" target="_self">Th30</a></i>;<br/>


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