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<div class="add">

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

<b>thus </b><a NAME="E1:2"/>
( <font color="Maroon">AFP</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a>  implies for <font color="Olive">o</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><br/> for <font color="Olive">K</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><br/> for <font color="Olive">c</font>, <font color="Olive">c'</font>, <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">c'</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">b</font>,<font color="Olive">b'</font> &amp; not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a> <font color="Olive">K</font> &amp; <font color="Olive">a'</font>,<font color="Olive">b'</font> <a href="aff_1.html#R2">//</a> <font color="Olive">K</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">c'</font> holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b'</font>,<font color="Olive">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">AFP</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</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">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c'</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>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>assume </b><a NAME="E2:2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 ;<br/>

<a NAME="E3:2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E3:2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/><a NAME="E4:2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E5:2_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E6:2_1_1"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c'</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E7:2_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E8:2_1_1"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b'</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E9:2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E10:2_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> implies <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/><b>hence </b><a NAME="E11:2_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:2_1_2_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</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> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E3:2_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/><a NAME="E4:2_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_2_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a'</font>,<font color="Maroon">o</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 ;<br/><a NAME="E2:2_1_2_1_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:2_1_2_1_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E4:2_1_2_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> or <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_2_1_1_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/><b>hence </b><a NAME="E3:2_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/></div><b>end;</b></div><a NAME="E6:2_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<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#T37">AFF_1:37</a></i>;<br/><a NAME="E7:2_1_2_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b'</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E8:2_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:2_1_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_2.html#D7">AFF_2:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">o</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><br/> for <font color="Olive">K</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><br/> for <font color="Olive">c</font>, <font color="Olive">c'</font>, <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">c'</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">b</font>,<font color="Olive">b'</font> &amp; not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">K</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a> <font color="Olive">K</font> &amp; <font color="Olive">a'</font>,<font color="Olive">b'</font> <a href="aff_1.html#R2">//</a> <font color="Olive">K</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">c'</font> holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b'</font>,<font color="Olive">c'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><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>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>let </b><font color="Maroon">c</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>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</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">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E2:2_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E3:2_2"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2"/>
<font color="Maroon">AFP</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D7">AFF_2:def 7</a></i>;<br/>


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