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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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font>;<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="E1:108"/><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="E2:108"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">g</font> <a href="transgeo.html#NR9" target="_self">is_Tr</a> 
 <b>and </b><br/><a NAME="E3:108"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E4:108"/><i><font color="Green">E28</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/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:108_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFS</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFS</font> )
 <b>by </b><i>, , , <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E3:108_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:108_2"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_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">AFS</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFS</font><b> such that </b><br/><a NAME="E3:108_2_1_1_1"/><i><font color="Green">E35</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="aff_1.html#T22">AFF_1:22</a></i>;<br/><a NAME="E4:108_2_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , <a class="ref" href="transgeo.html#D7" target="_self">Def7</a>, </i>;<br/><a NAME="E5:108_2_1_1_1"/>
<font color="Maroon">f</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>
 <b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E6:108_2_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="transgeo.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E7:108_2_1_1_1"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:108_2_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:108_2_1_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E3:108_2_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E4:108_2_1_1_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<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">x</font>
 
<b>by </b><i>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a>, </i>;<br/>
<a NAME="E5:108_2_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E6:108_2_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E7:108_2_1_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<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">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E8:108_2_1_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<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">a</font>
 
<b>by </b><i><a class="ref" href="transgeo.html#D6" target="_self">Def6</a>, <a class="ref" href="transgeo.html#T42" target="_self">Th42</a>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<b>hence </b><a NAME="E9:108_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="transgeo.html#D7" target="_self">Def7</a>, <a class="ref" href="transgeo.html#D8" target="_self">Def8</a>, <a class="ref" href="transgeo.html#T51" target="_self">Th51</a>, <a class="ref" href="transgeo.html#T56" target="_self">Th56</a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:108_2_1_1_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">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="transgeo.html#T44" target="_self">Th44</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:108_2_1_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">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:108_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">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:108_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:108"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T40" target="_self">Th40</a></i>;<br/>


</div>
