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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="E1:93"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 <b>and </b><br/><a NAME="E2:93"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">g</font> <a href="transgeo.html#NR8" target="_self">is_Dil</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/><b>set </b><font color="Maroon">x'</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><b>set </b><font color="Maroon">y'</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>;<br/><a NAME="E1:93_1"/><i><font color="Green">E26</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">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>
 <b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/><a NAME="E2:93_1"/><i><font color="Green">E28</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">y</font> <a href="hidden.html#R1">=</a> <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">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>;<br/><a NAME="E3:93_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <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>,<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">y</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="transgeo.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:93_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T85">FUNCT_2:85</a></i>;<br/><b>hence </b><a NAME="E3:93_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <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>,<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">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:93_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <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>,<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">y</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E43"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="transgeo.html#E44"><i><font color="Green">Lemma63</font></i></a>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:93"/>
<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#D4" target="_self">Def4</a></i>;<br/>


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