<?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="E1:21"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_3.html#NR3" title="AFF_3:NR.3">satisfies_DES1_2</a> 
 ;<br/>

<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_3.html#D4" target="_self" title="AFF_3:def.4">AFF_3:def 4</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">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">c'</font>, <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</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><b>that </b><br/><a NAME="E2:21"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">P</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="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> )
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">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="c6">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="c7">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">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="c5">o</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="c11">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="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</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="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c8">b</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="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a'</font> )
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green" title="E13">A5</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font> )
 ;<br/>

<b>assume </b><a NAME="E6:21"/><i><font color="Green" title="E14">A6</font></i>: 
not <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E7:21"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c7">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="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E8:21"/><i><font color="Green" title="E16">A8</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon" title="c6">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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</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="E3:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</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="E9:21"/><i><font color="Green" title="E17">A9</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="c11">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E2:21_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E8:21"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:21"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E11:21"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c12">p</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="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c13">q</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="21_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E20">A12</font></i>: 
(  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> or  not <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> or  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> or  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> )
 ;<br/>

<div><i><font color="Green" title="E21">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 ;<br/><a NAME="E2:21_3_1"/>
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E3:21_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E1:21_3_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3_2"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 ;<br/><a NAME="E2:21_3_2"/>
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E10:21"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E3:21_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E1:21_3_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E10:21"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E1:21_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="aff_3.html#E2:21_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:21"/><i><font color="Green" title="E20">A16</font></i>: 
(  not <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_4"/>
( <font color="Maroon" title="c3">P</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="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> )
 ;<br/>

<a NAME="E2:21_4"/><b>then </b>
<font color="Maroon" title="c4">C</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="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:21_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">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>set </b><font color="Maroon" title="c14">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>;<br/>
<b>set </b><font color="Maroon" title="c15">T</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>;<br/>
<b>set </b><font color="Maroon" title="c16">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>;<br/>
<b>set </b><font color="Maroon" title="c17">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font>;<br/>
<b>set </b><font color="Maroon" title="c18">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>;<br/>
<a NAME="E13:21"/><i><font color="Green" title="E21">A17</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</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="c8">b</font>,<font color="Maroon" title="c6">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="c12">p</font>,<font color="Maroon" title="c13">q</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="c7">a'</font>,<font color="Maroon" title="c11">c'</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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">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="c12">p</font>,<font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c13">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="c12">p</font>,<font color="Maroon" title="c13">q</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="c8">b</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">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="c8">b</font>,<font color="Maroon" title="c10">c</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="c8">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c6">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">b</font>,<font color="Maroon" title="c6">a</font> &amp; <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="c7">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c11">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="c7">a'</font>,<font color="Maroon" title="c11">c'</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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E14:21"/><b>then </b><i><font color="Green" title="E22">A18</font></i>: 
( <font color="Maroon" title="c13">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">b</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c12">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="c8">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c12">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="c9">b'</font>,<font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<div><i><font color="Green" title="E23">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_5"/>
not <font color="Maroon" title="c3">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">x</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="E3:21_5"/><i><font color="Green" title="E24">A20</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c19">x</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_3.html#E2:21"><i><font color="Green" title="E10">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="E4:21_5"/><i><font color="Green" title="E25">A21</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5_1"/><i><font color="Green" title="E26">A22</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ;<br/>

<a NAME="E2:21_5_1"/><b>then </b><i><font color="Green" title="E27">A23</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c6">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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E3:21_5_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E13:21"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="aff_3.html#E1:21_5_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:21_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E11:21"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E13:21"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="aff_3.html#E14:21"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="aff_3.html#E2:21_5_1"><i><font color="Green" title="E27">A23</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="E5:21_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5_2"/><i><font color="Green" title="E26">A24</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ;<br/>

<a NAME="E2:21_5_2"/><b>then </b>
<font color="Maroon" title="c12">p</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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E3:21_5_2"/><b>then </b>
( <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c13">q</font> or <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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E10:21"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E13:21"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="aff_3.html#E14:21"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E4:21_5_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="aff_3.html#E1:21_5_2"><i><font color="Green" title="E26">A24</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#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:21_5_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</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="c13">q</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</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="E6:21_5_2"/><b>then </b>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E7:21_5_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E8:21_5_2"/><b>then </b>
<font color="Maroon" title="c8">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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E9:21"><i><font color="Green" title="E17">A9</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="E9:21_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="aff_3.html#E1:21_5_2"><i><font color="Green" title="E26">A24</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="E6:21_5"/><b>then </b>
<font color="Maroon" title="c19">x</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_3.html#E1:21"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E9:21"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="aff_3.html#E4:21_5"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E7:21_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E6:21"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:21_5"><i><font color="Green" title="E24">A20</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_6"/>
not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">x</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="E3:21_6"/><i><font color="Green" title="E24">A25</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c19">x</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_3.html#E2:21"><i><font color="Green" title="E10">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="E4:21_6"/><i><font color="Green" title="E25">A26</font></i>: 
<font color="Maroon" title="c19">x</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="21_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_6_1"/><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 ;<br/>

<a NAME="E2:21_6_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c9">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:21_6_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">b'</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="c13">q</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</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#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:21_6_1"/><b>then </b><i><font color="Green" title="E27">A28</font></i>: 
<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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E9:21"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E11:21"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E5:21_6_1"/><b>then </b><i><font color="Green" title="E28">A29</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="aff_3.html#E1:21_6_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E6:21_6_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="aff_3.html#E1:21_6_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="aff_3.html#E4:21_6_1"><i><font color="Green" title="E27">A28</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E7:21_6_1"/><b>then </b>
<font color="Maroon" title="c13">q</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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E9:21"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E8:21_6_1"/><b>then </b>
<font color="Maroon" title="c4">C</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="c8">b</font>,<font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E11:21"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="aff_3.html#E13:21"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="aff_3.html#E14:21"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="aff_3.html#E5:21_6_1"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E9:21_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E13:21"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="aff_3.html#E4:21_6_1"><i><font color="Green" title="E27">A28</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="E5:21_6"/><i><font color="Green" title="E26">A30</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_6_2"/><i><font color="Green" title="E27">A31</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ;<br/>

<a NAME="E2:21_6_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:21_6_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E7:21"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E4:21_6_2"/><b>then </b>
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E9:21"><i><font color="Green" title="E17">A9</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#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E5:21_6_2"/><b>then </b>
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font> or <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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E6:21_6_2"/><b>then </b>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E10:21"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="aff_3.html#E1:21_6_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E7:21_6_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c12">p</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="E8:21_6_2"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</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="E9:21_6_2"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></i>;<br/>
<a NAME="E10:21_6_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E11:21_6_2"/><b>then </b>
<font color="Maroon" title="c8">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_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</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="E12:21_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="aff_3.html#E1:21_6_2"><i><font color="Green" title="E27">A31</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="E6:21_6"/><i><font color="Green" title="E27">A32</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E7:21_6"/>
( <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E8:21_6"/><b>then </b>
<font color="Maroon" title="c19">x</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_3.html#E1:21"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:21"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="aff_3.html#E4:21_6"><i><font color="Green" title="E25">A26</font></i></a>, <a class="txt" href="aff_3.html#E5:21_6"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="aff_3.html#E6:21_6"><i><font color="Green" title="E27">A32</font></i></a>, <a class="ref" href="aff_3.html#D3" target="_self" title="AFF_3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E9:21_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:21"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:21"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:21"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E6:21"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E3:21_6"><i><font color="Green" title="E24">A25</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>
<b>hence </b><a NAME="E17:21"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E12:21"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="aff_3.html#E15:21"><i><font color="Green" title="E23">A19</font></i></a></i>;<br/>


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