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

<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>] <b>means </b>( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a<sub>2</sub></font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">p'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Olive">p</font>,<font color="Olive">p'</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">p'</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">a<sub>2</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a<sub>2</sub></font> ) ) );<br/>
<b>assume </b><a NAME="E1:15"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">AFP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:15"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:15"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>,<font color="Olive">y</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:15"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">s</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/>
<a NAME="E6:15"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> st <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<b>from </b><i><a class="ref" href="transgeo.html#S1">TRANSGEO:sch 1</a>(, , , );<br/></i>
<b>hence </b><a NAME="E7:15"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AFP</font> st <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> iff ( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Olive">x</font>,<font color="Olive">y</font> &amp; <font color="Maroon">a</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">y</font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">x</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">p'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Olive">p</font>,<font color="Olive">p'</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">p'</font> &amp; <font color="Olive">p</font>,<font color="Olive">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Olive">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">y</font> ) ) ) )
 ;<br/>


</div>
