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<div class="add">

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

<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a> 
 ;<br/>

<b>thus </b><a NAME="E2:40"/>
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR7" title="AFF_2:NR.7">satisfies_TDES</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D7" target="_self" title="AFF_2:def.7">AFF_2:def 7</a><br/><b>let </b><font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b'</font>, <font color="Maroon" title="c9">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>;<br/>















<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c2">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c9">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> )
 ;<br/>

<div><i><font color="Green" title="E20">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ;<br/><a NAME="E2:40_1_1"/><b>then </b>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c10">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>;<br/>
<b>set </b><font color="Maroon" title="c11">P</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>;<br/>
<a NAME="E3:40_1"/><i><font color="Green" title="E21">A4</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:40_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E4:40_1"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:40_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E5:40_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_2"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c5">b</font>
 ;<br/>

<a NAME="E2:40_1_2"/><b>then </b>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:40_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/>
<a NAME="E3:40_1_2"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:40_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E4:40_1_2"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c4">a</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E3:40_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:40_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c9">c'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:40"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="aff_2.html#E1:40_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:40_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E3:40_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E4:40_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="aff_2.html#D4" target="_self" title="AFF_2:def.4">Def4</a></i>;<br/>


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


</div>
