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<b>let </b><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> holds <br/> ex <font color="Olive" title="b4">f</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">AFP</font> st <br/>( <font color="Olive" title="b4">f</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a>  &amp; <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> &amp; <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">q'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/><b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E8">A2</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">p'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c8">q'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c6">p'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c8">q'</font> )
 ;<br/><div><i><font color="Green" title="E9">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font>
 ;<br/><a NAME="E2:6_1_1"/><b>then </b>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/><a NAME="E3:6_1_1"/><b>then </b>
<font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="homothet.html#E1:6_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/><b>hence </b><a NAME="E4:6_1_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a'</font>
 ;<br/><a NAME="E2:6_1_2"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">f</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">AFP</font><b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="transgeo.html#NR8" title="TRANSGEO:NR.8">is_Dil</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="homothet.html#E1:6_1_2"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font>;<br/><b>set </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font>;<br/><a NAME="E5:6_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_2_1"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E4:6_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="transgeo.html#T85" title="TRANSGEO:th.85">TRANSGEO:85</a></i>;<br/>
<a NAME="E2:6_1_2_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_2_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:6_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_2_2"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E4:6_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="transgeo.html#T85" title="TRANSGEO:th.85">TRANSGEO:85</a></i>;<br/>
<a NAME="E2:6_1_2_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_2_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="aff_1.html#T70" title="AFF_1:th.70">AFF_1:70</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:6_1_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:6_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="homothet.html#E5:6_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="transgeo.html#T85" title="TRANSGEO:th.85">TRANSGEO:85</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:6_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c8">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E2:6_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon" title="c1">AFP</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>


</div>
