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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">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>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">OAS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#NR4" target="_self">is_CDil</a> 
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">g</font> <a href="transgeo.html#NR5" target="_self">is_SDil</a> 
 ;<br/>

<b>thus </b><a NAME="E3:48"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> holds  <font color="Olive">x</font>,<font color="Olive">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="Olive">y</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="Olive">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="transgeo.html#D7" target="_self">TRANSGEO:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </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">OAS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</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="E1:48_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> &amp; <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="E2:48_1"/><i><font color="Green">E28</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">y</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">y</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">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:48_1"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_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="E2:48_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="E3:48_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">y</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">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_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">y</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">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E2:48_1"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="transgeo.html#E3:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:48"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="transgeo.html#D7" target="_self">TRANSGEO:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </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">OAS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>



<b>set </b><font color="Maroon">x'</font> = <font color="Maroon">f</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>;<br/>
<a NAME="E1:48_2"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</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="E2:48_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon">f</font> <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> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:48_2_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="E3:48_2_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">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_2"/>
<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">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="transgeo.html#K1" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/>


</div><b>end;</b></div>


</div>
