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<b>let </b><font color="Maroon" title="c1">AFS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font>  st <font color="Olive" title="b1">f</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  &amp; <font color="Olive" title="b2">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> </span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  implies <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:110"/><i><font color="Green" title="E72">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> 
 <b>and </b><br/><a NAME="E2:110"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> </span><br/>

<a NAME="E3:110"/><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:110"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>;<br/>
<a NAME="E4:110"/>
<font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:110"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>;<br/>
<a NAME="E5:110"/><b>then </b><i><font color="Green" title="E75">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E3:110"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="transgeo.html#T88" target="_self" title="TRANSGEO:th.88">Th88</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1"/><i><font color="Green" title="E76">A5</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> </span><br/><a NAME="E2:110_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font>  holds <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font><b> such that </b><a NAME="E2:110_1_1"/><i><font color="Green" title="E77">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:110_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:110_1_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>;<br/>
<a NAME="E4:110_1_1"/><b>then </b>
( <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a>  )
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:110"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a>, <a class="ref" href="transgeo.html#T104" target="_self" title="TRANSGEO:th.104">Th104</a></i>;<br/>
<a NAME="E5:110_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:110"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="transgeo.html#T103" target="_self" title="TRANSGEO:th.103">Th103</a></i>;<br/>
<b>hence </b><a NAME="E6:110_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E1:110_1"><i><font color="Green" title="E76">A5</font></i></a>, <a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:110_1"/>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> 
 <b>by </b><i><a class="txt" href="transgeo.html#E5:110"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:110"/>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR9" title="TRANSGEO:NR.9">is_Tr</a> 
 <b>by </b><i><a class="txt" href="transgeo.html#E5:110"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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