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













<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">AFP</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a>  )
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E44</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">x</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E45</font></i>: 
( ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">r</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">r</font>,<font color="Maroon">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 ;<br/>

<a NAME="E4:30"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T48">AFF_1:48</a>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green">E47</font></i>: 
( ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">y</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:30"/>
( ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> ) or ( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">y</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">r</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:30"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , </i>;<br/>


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