<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">AFS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font>;<br/><b>let </b><font color="Maroon" title="c4">f</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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:100"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a> 
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green" title="E67">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font>;<br/><a NAME="E1:100_1"/><i><font color="Green" title="E68">A3</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E1:100"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="transgeo.html#E2:100"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="transgeo.html#T94" target="_self" title="TRANSGEO:th.94">Th94</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_1_1"/><i><font color="Green" title="E69">A4</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_1_1_1"/><i><font color="Green" title="E70">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">z</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><br/><a NAME="E3:100_1_1_1"/><i><font color="Green" title="E71">A6</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E2:100"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="aff_1.html#T22" title="AFF_1:th.22">AFF_1:22</a></i>;<br/><a NAME="E4:100_1_1_1"/><i><font color="Green" title="E72">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E1:100"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="transgeo.html#E2:100"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="transgeo.html#E3:100_1_1_1"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="transgeo.html#T94" target="_self" title="TRANSGEO:th.94">Th94</a></i>;<br/><a NAME="E5:100_1_1_1"/>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E2:100_1_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:100_1_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:100_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E1:100_1_1_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="transgeo.html#E3:100_1_1_1"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:100_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E1:100"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="transgeo.html#E2:100"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="transgeo.html#E4:100_1_1_1"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="transgeo.html#T94" target="_self" title="TRANSGEO:th.94">Th94</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:100_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E2:100"><i><font color="Green" title="E67">A2</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:100_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E1:100_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:100"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
