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<div class="add">

<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>thus </b><a NAME="E1:113"/>
( <font color="Maroon">f</font> <a href="transgeo.html#NR10" target="_self">is_Col</a>  implies for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> holds <br/> ( <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">z</font>,<font color="Olive">t</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:113_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#R2" target="_self">is_automorphism_of</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="transgeo.html#D12" target="_self">TRANSGEO:def 12</a><br/>

<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>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/>







<b>thus </b><a NAME="E2:113_1"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> implies <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:113_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:113_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>
<a NAME="E3:113_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:113_1_1"/>
<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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E3:113_1"/>
<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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E4:113_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>
<a NAME="E5:113_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:113_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:113"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> holds <br/> ( <font color="Olive">x</font>,<font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Olive">z</font>,<font color="Olive">t</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="transgeo.html#D3" target="_self">TRANSGEO:def 3</a>,<a class="ref" href="transgeo.html#D12" target="_self">TRANSGEO:def 12</a><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>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<br/>







<a NAME="E3:113"/>
( <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> iff <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:113"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AFS</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>


</div>
