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<b>let </b><font color="Maroon">AFS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<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">AFS</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:110"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E2:110"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 ;<br/>

<a NAME="E3:110"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 
<b>by </b><i>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E4:110"/>
<font color="Maroon">g</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 
<b>by </b><i>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E5:110"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 
<b>by </b><i>, <a class="ref" href="transgeo.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFS</font>
 ;<br/><a NAME="E2:110_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>  holds <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font><b> such that </b><a NAME="E1:110_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:110_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#T42" target="_self">Th42</a>, <a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/>
<a NAME="E3:110_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K2">"</a>  <a href="transgeo.html#NR9" target="_self">is_Tr</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:110_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:110_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:110_1"/>
<font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#T40" target="_self">Th40</a>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:110"/>
<font color="Maroon">f</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#T40" target="_self">Th40</a>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/>


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