<?xml version="1.0"?>
<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="E1:41"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR8" title="AFF_2:NR.8">satisfies_TDES_1</a> 
 ;<br/>

<b>thus </b><a NAME="E2:41"/>
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR12" title="AFF_2:NR.12">satisfies_des_1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</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#D12" target="_self" title="AFF_2:def.12">AFF_2:def 12</a><br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">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="E1:41_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">c'</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font> )
 ;<br/>

<b>assume </b><a NAME="E2:41_1"/><i><font color="Green" title="E21">A3</font></i>: 
not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">o</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</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">AP</font><b> such that </b><br/><a NAME="E6:41_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c2">A</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="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E8:41_1"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">N</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E9:41_1"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c13">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c12">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c13">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E10:41_1"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c13">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c12">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c13">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E11:41_1"/><i><font color="Green" title="E27">A9</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E12:41_1"/><i><font color="Green" title="E28">A10</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E13:41_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font>
 ;<br/>

<a NAME="E2:41_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:41_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E5:41_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:41_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:41_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c6">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:41_1"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 ;<br/>

<a NAME="E2:41_1_2"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:41_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a'</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="E4:41_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E5:41_1_2"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E13:41_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</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>
<a NAME="E15:41_1"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_3"/>
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c'</font>
 ;<br/>

<a NAME="E2:41_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:41_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E12:41_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:41_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E5:41_1_3"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</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/>
<b>hence </b><a NAME="E6:41_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c14">P'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/>
<b>set </b><font color="Maroon" title="c15">C'</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>;<br/>
<a NAME="E16:41_1"/><i><font color="Green" title="E32">A15</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">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="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">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="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c9">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="c8">a'</font>,<font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c8">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="c8">a'</font>,<font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E11:41_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="aff_2.html#E12:41_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E17:41_1"/><i><font color="Green" title="E33">A16</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c8">a'</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E11:41_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="aff_2.html#E12:41_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="aff_1.html#T51" title="AFF_1:th.51">AFF_1:51</a></i>;<br/>
<a NAME="E18:41_1"/>
not <font color="Maroon" title="c12">M</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="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_4"/>
<font color="Maroon" title="c12">M</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="c5">a</font>,<font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E2:41_1_4"/><b>then </b>
<font color="Maroon" title="c2">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="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:41_1_4"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</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>then consider </b><font color="Maroon" title="c16">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E20:41_1"/><i><font color="Green" title="E34">A17</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c16">p</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="c5">a</font>,<font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E9:41_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E21:41_1"/>
not <font color="Maroon" title="c13">N</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="c8">a'</font>,<font color="Maroon" title="c9">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_5"/>
<font color="Maroon" title="c13">N</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="c8">a'</font>,<font color="Maroon" title="c9">b'</font>
 ;<br/>

<a NAME="E2:41_1_5"/><b>then </b>
<font color="Maroon" title="c2">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="c8">a'</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:41_1_5"/><b>then </b>
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</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>then consider </b><font color="Maroon" title="c17">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E23:41_1"/><i><font color="Green" title="E35">A18</font></i>: 
( <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">N</font> &amp; <font color="Maroon" title="c17">q</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="c8">a'</font>,<font color="Maroon" title="c9">b'</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E9:41_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E24:41_1"/>
not <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:41_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><b> such that </b><br/><a NAME="E26:41_1"/><i><font color="Green" title="E36">A19</font></i>: 
( <font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c18">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E27:41_1"/><i><font color="Green" title="E37">A20</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E26:41_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E28:41_1"/><i><font color="Green" title="E38">A21</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E29:41_1"/><i><font color="Green" title="E39">A22</font></i>: 
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c16">p</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E10:41_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E30:41_1"/><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="aff_2.html#E17:41_1"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E31:41_1"/><i><font color="Green" title="E41">A24</font></i>: 
not <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_6"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E2:41_1_6"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E11:41_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:41_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:41_1"/><i><font color="Green" title="E42">A25</font></i>: 
<font color="Maroon" title="c18">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_7"/>
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E2:41_1_7"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E26:41_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:41_1_7"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E11:41_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:41_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="aff_2.html#E14:41_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:41_1"/><i><font color="Green" title="E43">A26</font></i>: 
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c16">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c17">q</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="aff_2.html#E10:41_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E34:41_1"/>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E35:41_1"/><b>then </b><i><font color="Green" title="E44">A27</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c17">q</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E26:41_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="aff_2.html#E27:41_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="aff_2.html#E28:41_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="aff_2.html#E29:41_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="aff_2.html#E30:41_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="aff_2.html#E31:41_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="aff_2.html#E32:41_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="txt" href="aff_2.html#E33:41_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="aff_2.html#D8" target="_self" title="AFF_2:def.8">Def8</a></i>;<br/>
<a NAME="E36:41_1"/><i><font color="Green" title="E45">A28</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E4:41_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E37:41_1"/><i><font color="Green" title="E46">A29</font></i>: 
not <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_8"/>
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:41_1_8"/><b>then </b>
( <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:41_1_8"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E15:41_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:41_1"/><i><font color="Green" title="E47">A30</font></i>: 
<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c17">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="aff_2.html#E10:41_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E39:41_1"/><i><font color="Green" title="E48">A31</font></i>: 
<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E40:41_1"/><i><font color="Green" title="E49">A32</font></i>: 
<font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c8">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="aff_2.html#E17:41_1"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E41:41_1"/><i><font color="Green" title="E50">A33</font></i>: 
<font color="Maroon" title="c10">c'</font>,<font color="Maroon" title="c17">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E42:41_1"/><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_9"/>
<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font>
 ;<br/>

<a NAME="E2:41_1_9"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:41_1_9"/><b>then </b>
<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">c'</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="E4:41_1_9"/><b>then </b>
<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E15:41_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E5:41_1_9"/><b>then </b>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E12:41_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:41_1"/>
<font color="Maroon" title="c11">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_10"/>
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font>
 ;<br/>

<a NAME="E2:41_1_10"/><b>then </b>
( <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E4:41_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:41_1_10"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E15:41_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="aff_1.html#T62" title="AFF_1:th.62">AFF_1:62</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E13:41_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:41_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E4:41_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="aff_2.html#E36:41_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="aff_2.html#E37:41_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="aff_2.html#E38:41_1"><i><font color="Green" title="E47">A30</font></i></a>, <a class="txt" href="aff_2.html#E39:41_1"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="aff_2.html#E40:41_1"><i><font color="Green" title="E49">A32</font></i></a>, <a class="txt" href="aff_2.html#E41:41_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="aff_2.html#E42:41_1"><i><font color="Green" title="E51">A34</font></i></a>, <a class="ref" href="aff_2.html#D8" target="_self" title="AFF_2:def.8">Def8</a></i>;<br/>
<a NAME="E45:41_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c11">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c18">s</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c17">q</font>,<font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E35:41_1"><i><font color="Green" title="E44">A27</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E46:41_1"/><b>then </b><i><font color="Green" title="E52">A35</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_11"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_11"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font>
 ;<br/><a NAME="E2:41_1_11"/><b>then </b>
<font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">N</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E10:41_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E23:41_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><b>hence </b><a NAME="E3:41_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:41_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E8:41_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="aff_2.html#E9:41_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E48:41_1"/><b>then </b>
( <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s</font> or <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E4:41_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="aff_2.html#E26:41_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="aff_2.html#E46:41_1"><i><font color="Green" title="E52">A35</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E49:41_1"/><b>then </b>
<font color="Maroon" title="c7">c</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="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E2:41_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:41_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="aff_2.html#E6:41_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_2.html#E9:41_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="aff_2.html#E20:41_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="aff_2.html#E26:41_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E50:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_2.html#E16:41_1"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>


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


</div>
