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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>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">q</font>, <font color="Maroon" title="c9">q'</font>, <font color="Maroon" title="c10">s</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>let </b><font color="Maroon" title="c11">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> )
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c1">AFP</font> <a href="aff_2.html#NR7" title="AFF_2:NR.7">satisfies_TDES</a> 
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font> )
 <b>and </b><br/><a NAME="E4:29"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c9">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">s</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c10">s</font> )
 ;<br/>

<b>assume </b><a NAME="E5:29"/><i><font color="Green" title="E34">A5</font></i>: 
not <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<a NAME="E6:29"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:29"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="homothet.html#E2:29"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="homothet.html#E3:29"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="homothet.html#E4:29"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="homothet.html#E29"><i><font color="Green" title="E28">Lm21</font></i></a></i>;<br/>
<a NAME="E7:29"/><i><font color="Green" title="E36">A7</font></i>: 
( not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">K</font> &amp; <font color="Maroon" title="c11">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:29"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c10">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E4:29"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="homothet.html#E6:29"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c7">y</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/>
<a NAME="E10:29"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">q'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E12:29"/><i><font color="Green" title="E38">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c11">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">M</font> )
 <b>by </b><i><a class="txt" href="homothet.html#E7:29"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E13:29"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c12">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E14:29"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c12">M</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:29"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E15:29"/><b>then </b><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E16:29"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">s</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c12">M</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E4:29"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>
<a NAME="E18:29"/><b>then </b>
<font color="Maroon" title="c12">M</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="c7">y</font>,<font color="Maroon" title="c10">s</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E5:29"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="homothet.html#E13:29"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="homothet.html#E15:29"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E19:29"/><b>then </b>
<font color="Maroon" title="c9">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E10:29"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:29"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:29"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="homothet.html#E4:29"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="homothet.html#E12:29"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


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