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<b>let </b><font color="Maroon">AS</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="analoaf.html#L1">AffinStruct</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">AS</font>;<br/>



<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">f</font> <a href="transgeo.html#R3" target="_self">is_DIL_of</a> <font color="Maroon">AS</font>
 ;<br/><a NAME="E2:28_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> <a href="transgeo.html#R1" target="_self">is_FormalIz_of</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AS</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><a NAME="E3:28_1"/>
<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">a</font>,<font color="Maroon">b</font></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">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</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">AS</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds  <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>
 ;<br/><a NAME="E2:28_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds  <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="Olive">x</font>,<font color="Olive">y</font></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="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</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">AS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_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">AS</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>



<a NAME="E1:28_2_1"/>
<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/>;<br/>
<b>hence </b><a NAME="E2:28_2_1"/>
<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"><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><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">AS</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:28_2"/><b>then </b>
<font color="Maroon">f</font> <a href="transgeo.html#R1" target="_self">is_FormalIz_of</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AS</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:28_2"/>
<font color="Maroon">f</font> <a href="transgeo.html#R3" target="_self">is_DIL_of</a> <font color="Maroon">AS</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
( <font color="Maroon">f</font> <a href="transgeo.html#R3" target="_self">is_DIL_of</a> <font color="Maroon">AS</font> iff for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> holds  <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> )
 <b>by </b><i/>;<br/>


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