<?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="E1:39"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR2" title="AFF_2:NR.2">satisfies_PAP</a> 
 ;<br/>

<a NAME="E2:39"/><b>then </b>
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR13" title="AFF_2:NR.13">satisfies_pap</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T23" target="_self" title="AFF_2:th.23">Th23</a></i>;<br/>
<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR1" title="AFF_2:NR.1">satisfies_PPAP</a> 
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="aff_2.html#T24" target="_self" title="AFF_2:th.24">Th24</a></i>;<br/>
<b>thus </b><a NAME="E4:39"/>
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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#D4" target="_self" title="AFF_2:def.4">AFF_2: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">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">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="E1:39_1"/><i><font color="Green" title="E20">A3</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="c5">o</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="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="c7">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="c8">c</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="c9">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">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="c10">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="c8">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="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> &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="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 ;<br/>

<b>assume </b><a NAME="E2:39_1"/><i><font color="Green" title="E21">A4</font></i>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E3:39_1"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c10">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="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/>
<a NAME="E4:39_1"/><i><font color="Green" title="E23">A6</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="c7">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="c8">c</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E5:39_1"/><i><font color="Green" title="E24">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="c9">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ;<br/>

<a NAME="E2:39_1_1"/><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">b</font>,<font color="Maroon" title="c10">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</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="E3:39_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c6">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c6">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="E4:39_1_1"/><b>then </b>
( ( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font> or <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> ) &amp; ( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font> or <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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</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="E6:39_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 ;<br/>

<a NAME="E2:39_1_2"/><b>then </b>
( <font color="Maroon" title="c9">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</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="E3:39_1_2"/><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> or <font color="Maroon" title="c9">a'</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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</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="E4:39_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</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="E7:39_1"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c9">a'</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="39_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_3"/>
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E2:39_1_3"/><b>then </b>
( <font color="Maroon" title="c9">a'</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">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</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="E3:39_1_3"/><b>then </b>
( <font color="Maroon" title="c9">a'</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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</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="E4:39_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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="E8:39_1"/><i><font color="Green" title="E27">A11</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_4"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E2:39_1_4"/><b>then </b>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c9">a'</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#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:39_1_4"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:39_1_4"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_4"/><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">b</font>,<font color="Maroon" title="c8">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="E6:39_1_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">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="E7:39_1_4"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">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="E8:39_1_4"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="E9:39_1_4"/><b>then </b>
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="aff_2.html#E3:39_1_4"><i><font color="Green" title="E28">A12</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#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:39_1"/><i><font color="Green" title="E28">A13</font></i>: 
not  <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">b</font>,<font color="Maroon" title="c8">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_5"/>
 <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">b</font>,<font color="Maroon" title="c8">c</font>
 ;<br/>

<a NAME="E2:39_1_5"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">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="E3:39_1_5"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:39_1_5"/><b>then </b>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</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="E5:39_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E8:39_1"><i><font color="Green" title="E27">A11</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="E10:39_1"/><i><font color="Green" title="E29">A14</font></i>: 
not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_6"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ;<br/>

<a NAME="E2:39_1_6"/><b>then </b>
( <font color="Maroon" title="c7">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="c10">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#E1:39_1"><i><font color="Green" title="E20">A3</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="E3:39_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:39_1"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_7"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ;<br/>

<a NAME="E2:39_1_7"/><b>then </b>
( <font color="Maroon" title="c9">a'</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">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:39_1_7"/><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="c3">P</font> &amp; <font color="Maroon" title="c9">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</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="E4:39_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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="E12:39_1"/><i><font color="Green" title="E31">A16</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_8"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 ;<br/>

<a NAME="E2:39_1_8"/><b>then </b>
( <font color="Maroon" title="c10">b'</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">b'</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:39_1_8"/><b>then </b>
( <font color="Maroon" title="c7">b</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">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</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="E4:39_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_9"/><i><font color="Green" title="E32">A17</font></i>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>set </b><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>;<br/><b>set </b><font color="Maroon" title="c13">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>;<br/><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="c6">a</font>,<font color="Maroon" title="c8">c</font>;<br/><a NAME="E2:39_1_9"/><i><font color="Green" title="E33">A18</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c8">c</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c8">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">b</font>,<font color="Maroon" title="c8">c</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="c6">a</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">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="c6">a</font>,<font color="Maroon" title="c7">b</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="c6">a</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">K</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="E4:39_1_9"/><i><font color="Green" title="E34">A19</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><a NAME="E5:39_1_9"/><i><font color="Green" title="E35">A20</font></i>: 
( <font color="Maroon" title="c15">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c15">K</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="c7">b</font>,<font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E6:39_1_9"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 ;<br/>

<a NAME="E2:39_1_9_1"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">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="c7">b</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_9_1"/><b>then </b>
<font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">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="E4:39_1_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E9:39_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</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><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="E8:39_1_9"/><i><font color="Green" title="E36">A21</font></i>: 
( <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="c6">a</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1_9"><i><font color="Green" title="E35">A20</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="c17">T</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="E10:39_1_9"/><i><font color="Green" title="E37">A22</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><a NAME="E11:39_1_9"/><i><font color="Green" title="E38">A23</font></i>: 
( <font color="Maroon" title="c17">T</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c17">T</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="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E12:39_1_9"/>
not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_2"/>
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 ;<br/>

<a NAME="E2:39_1_9_2"/><b>then </b>
<font color="Maroon" title="c4">C</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="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_9_2"/><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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">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="E4:39_1_9_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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><b>then consider </b><font color="Maroon" title="c18">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="E14:39_1_9"/><i><font color="Green" title="E39">A24</font></i>: 
( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1_9"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><a NAME="E15:39_1_9"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_3"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 ;<br/>

<a NAME="E2:39_1_9_3"/><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">b</font> &amp;  <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="c9">a'</font> &amp;  <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="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:39_1_9_3"/><b>then </b>
<font color="Maroon" title="c7">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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="E16:39_1_9"/><i><font color="Green" title="E41">A26</font></i>: 
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> &amp; <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="aff_2.html#E10:39_1_9"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E17:39_1_9"/><b>then </b><i><font color="Green" title="E42">A27</font></i>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/><a NAME="E18:39_1_9"/><i><font color="Green" title="E43">A28</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_4"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ;<br/>

<a NAME="E2:39_1_9_4"/><b>then </b>
<font color="Maroon" title="c7">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</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="E3:39_1_9_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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="E19:39_1_9"/><i><font color="Green" title="E44">A29</font></i>: 
<font color="Maroon" title="c15">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1_9"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><b>set </b><font color="Maroon" title="c19">R</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font>;<br/><a NAME="E20:39_1_9"/><i><font color="Green" title="E45">A30</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c9">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">a'</font>,<font color="Maroon" title="c16">p</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="c9">a'</font>,<font color="Maroon" title="c16">p</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E18:39_1_9"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/><a NAME="E21:39_1_9"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_9_5"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font>
 ;<br/>

<a NAME="E2:39_1_9_5"/><b>then </b>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</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="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E10:39_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1_9"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/>
<a NAME="E3:39_1_9_5"/><b>then </b>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E4:39_1_9_5"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></i>;<br/>
<a NAME="E5:39_1_9_5"/><b>then </b>
<font color="Maroon" title="c16">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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green" title="E45">A30</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="E6:39_1_9_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1_9"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="aff_2.html#E19:39_1_9"><i><font color="Green" title="E44">A29</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>then consider </b><font color="Maroon" title="c20">r</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="E23:39_1_9"/><i><font color="Green" title="E46">A31</font></i>: 
( <font color="Maroon" title="c20">r</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">a'</font>,<font color="Maroon" title="c16">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/><a NAME="E24:39_1_9"/><i><font color="Green" title="E47">A32</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E25:39_1_9"/><i><font color="Green" title="E48">A33</font></i>: 
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_6"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_6"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_9_6"/><b>then </b>
<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c7">b</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="E3:39_1_9_6"/><b>then </b>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</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="E4:39_1_9_6"/><b>then </b><i><font color="Green" title="E49">A34</font></i>: 
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E10:39_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1_9"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_9_6"/>
 <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="c9">a'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E6:39_1_9_6"/><b>then </b>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">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="E7:39_1_9_6"/><b>then </b>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_9_6"/>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_9_6"><i><font color="Green" title="E49">A34</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:39_1_9"/><b>then </b><i><font color="Green" title="E49">A35</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green" title="E45">A30</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green" title="E46">A31</font></i></a>, <a class="txt" href="aff_2.html#E24:39_1_9"><i><font color="Green" title="E47">A32</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/><a NAME="E27:39_1_9"/><b>then </b><i><font color="Green" title="E50">A36</font></i>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E28:39_1_9"/><i><font color="Green" title="E51">A37</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_7"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_7"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E2:39_1_9_7"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c16">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="E3:39_1_9_7"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_9_7"/>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:39_1_9"/>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_8"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_9_8"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c18">q</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_9_8"/><b>then </b><i><font color="Green" title="E52">A38</font></i>: 
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_9_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_9_8_1"/>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 ;<br/><a NAME="E2:39_1_9_8_1"/><b>then </b>
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> &amp;  <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="c9">a'</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="aff_2.html#E8:39_1_9"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="aff_2.html#E14:39_1_9"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E3:39_1_9_8_1"/><b>then </b>
( <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <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="E4:39_1_9_8_1"/>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_9_8"/>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E25:39_1_9"><i><font color="Green" title="E48">A33</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_9_8"><i><font color="Green" title="E52">A38</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#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:39_1_9"/><b>then </b><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E20:39_1_9"><i><font color="Green" title="E45">A30</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1_9"><i><font color="Green" title="E46">A31</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1_9"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/><a NAME="E31:39_1_9"/><b>then </b>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="aff_2.html#E26:39_1_9"><i><font color="Green" title="E49">A35</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E32:39_1_9"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</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="E33:39_1_9"/><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="c10">b'</font>,<font color="Maroon" title="c20">r</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="E34:39_1_9"/><b>then </b>
<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c20">r</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="E35:39_1_9"/><b>then </b>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">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="E36:39_1_9"/><b>then </b>
<font color="Maroon" title="c20">r</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_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1_9"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E37:39_1_9"/><b>then </b>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E30:39_1_9"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E38:39_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1_9"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="aff_2.html#E16:39_1_9"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:39_1"/><b>then </b><i><font color="Green" title="E32">A40</font></i>: 
not <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>;<br/>
<b>set </b><font color="Maroon" title="c13">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>;<br/>
<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="c9">a'</font>,<font color="Maroon" title="c11">c'</font>;<br/>
<a NAME="E15:39_1"/><i><font color="Green" title="E33">A41</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</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">a'</font>,<font color="Maroon" title="c10">b'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c10">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">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; <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="c10">b'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c9">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">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c10">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">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c9">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">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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">K</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="E17:39_1"/><i><font color="Green" title="E34">A42</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E18:39_1"/><i><font color="Green" title="E35">A43</font></i>: 
( <font color="Maroon" title="c15">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c15">K</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="c10">b'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E19:39_1"/>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_10"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">K</font>
 ;<br/>

<a NAME="E2:39_1_10"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_10"/><b>then </b>
<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="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</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="E4:39_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E8:39_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</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>
<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="E21:39_1"/><i><font color="Green" title="E36">A44</font></i>: 
( <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="c9">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">K</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E18:39_1"><i><font color="Green" title="E35">A43</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="c17">T</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="E23:39_1"/><i><font color="Green" title="E37">A45</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E24:39_1"/><i><font color="Green" title="E38">A46</font></i>: 
( <font color="Maroon" title="c17">T</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c17">T</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="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E25:39_1"/>
not <font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_11"/>
<font color="Maroon" title="c4">C</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">T</font>
 ;<br/>

<a NAME="E2:39_1_11"/><b>then </b>
<font color="Maroon" title="c4">C</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="c9">a'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:39_1_11"/><b>then </b>
<font color="Maroon" title="c9">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</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="E4:39_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green" title="E30">A15</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>then consider </b><font color="Maroon" title="c18">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="E27:39_1"/><i><font color="Green" title="E39">A47</font></i>: 
( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">T</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E24:39_1"><i><font color="Green" title="E38">A46</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E28:39_1"/><i><font color="Green" title="E40">A48</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_12"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 ;<br/>

<a NAME="E2:39_1_12"/><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="c9">a'</font>,<font color="Maroon" title="c10">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c6">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c9">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E3:39_1_12"/><b>then </b>
( <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green" title="E30">A15</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#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E4:39_1_12"/><b>then </b>
( <font color="Maroon" title="c7">b</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">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</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="E5:39_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">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="E29:39_1"/><i><font color="Green" title="E41">A49</font></i>: 
( <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> &amp; <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="aff_2.html#E23:39_1"><i><font color="Green" title="E37">A45</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green" title="E39">A47</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E30:39_1"/><b>then </b><i><font color="Green" title="E42">A50</font></i>: 
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green" title="E39">A47</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/>
<a NAME="E31:39_1"/><i><font color="Green" title="E43">A51</font></i>: 
<font color="Maroon" title="c16">p</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="39_1_13"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_13"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 ;<br/>

<a NAME="E2:39_1_13"/><b>then </b>
<font color="Maroon" title="c10">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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E5:39_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</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="E3:39_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E12:39_1"><i><font color="Green" title="E31">A16</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="E32:39_1"/><i><font color="Green" title="E44">A52</font></i>: 
<font color="Maroon" title="c15">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E14:39_1"><i><font color="Green" title="E32">A40</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E33:39_1"/><i><font color="Green" title="E45">A53</font></i>: 
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_14"/>
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 ;<br/>

<a NAME="E2:39_1_14"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E12:39_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green" title="E39">A47</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="E3:39_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c19">R</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>;<br/>
<a NAME="E34:39_1"/><i><font color="Green" title="E46">A54</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &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="c6">a</font>,<font color="Maroon" title="c16">p</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="c6">a</font>,<font color="Maroon" title="c16">p</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E31:39_1"><i><font color="Green" title="E43">A51</font></i></a>, <a class="ref" href="aff_1.html#T38" title="AFF_1:th.38">AFF_1:38</a></i>;<br/>
<a NAME="E35:39_1"/>
not <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_15"/>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font>
 ;<br/>

<a NAME="E2:39_1_15"/><b>then </b>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</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="c2">A</font> )
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E30:39_1"><i><font color="Green" title="E42">A50</font></i></a>, <a class="txt" href="aff_2.html#E33:39_1"><i><font color="Green" title="E45">A53</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/>
<a NAME="E3:39_1_15"/><b>then </b>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</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#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E4:39_1_15"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c16">p</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#E11:39_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></i>;<br/>
<a NAME="E5:39_1_15"/><b>then </b>
<font color="Maroon" title="c16">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_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green" title="E46">A54</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="E6:39_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E17:39_1"><i><font color="Green" title="E34">A42</font></i></a>, <a class="txt" href="aff_2.html#E18:39_1"><i><font color="Green" title="E35">A43</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="aff_2.html#E32:39_1"><i><font color="Green" title="E44">A52</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>then consider </b><font color="Maroon" title="c20">r</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="E37:39_1"/><i><font color="Green" title="E47">A55</font></i>: 
( <font color="Maroon" title="c20">r</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="c6">a</font>,<font color="Maroon" title="c16">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="ref" href="aff_1.html#T73" title="AFF_1:th.73">AFF_1:73</a></i>;<br/>
<a NAME="E38:39_1"/><i><font color="Green" title="E48">A56</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E7:39_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E39:39_1"/><i><font color="Green" title="E49">A57</font></i>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_16"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_16"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green" title="E47">A55</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_16"/><b>then </b>
<font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">q</font>,<font color="Maroon" title="c10">b'</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="E3:39_1_16"/><b>then </b>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</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="E4:39_1_16"/><b>then </b><i><font color="Green" title="E50">A58</font></i>: 
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E30:39_1"><i><font color="Green" title="E42">A50</font></i></a>, <a class="txt" href="aff_2.html#E33:39_1"><i><font color="Green" title="E45">A53</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E5:39_1_16"/>
 <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="c9">a'</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E6:39_1_16"/><b>then </b>
<font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">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="E7:39_1_16"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_16"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="aff_2.html#E4:39_1_16"><i><font color="Green" title="E50">A58</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:39_1"/><b>then </b><i><font color="Green" title="E50">A59</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green" title="E39">A47</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green" title="E47">A55</font></i></a>, <a class="txt" href="aff_2.html#E38:39_1"><i><font color="Green" title="E48">A56</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/>
<a NAME="E41:39_1"/><b>then </b><i><font color="Green" title="E51">A60</font></i>: 
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E42:39_1"/><i><font color="Green" title="E52">A61</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_17"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_17"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c16">p</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E2:39_1_17"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c16">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="E3:39_1_17"/><b>then </b>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_17"/>
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E6:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:39_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_18"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_18"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c18">q</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green" title="E47">A55</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<a NAME="E2:39_1_18"/><b>then </b><i><font color="Green" title="E53">A62</font></i>: 
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c18">q</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_18_1"/>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q</font>
 ;<br/><a NAME="E2:39_1_18_1"/><b>then </b>
( <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c20">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c5">o</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E15:39_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="aff_2.html#E21:39_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="aff_2.html#E27:39_1"><i><font color="Green" title="E39">A47</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E3:39_1_18_1"/><b>then </b>
( <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font> &amp; <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <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="E4:39_1_18_1"/>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E11:39_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_18"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="aff_2.html#E39:39_1"><i><font color="Green" title="E49">A57</font></i></a>, <a class="txt" href="aff_2.html#E2:39_1_18"><i><font color="Green" title="E53">A62</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#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:39_1"/><b>then </b><i><font color="Green" title="E53">A63</font></i>: 
<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E3:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="aff_2.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="aff_2.html#E34:39_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="aff_2.html#E37:39_1"><i><font color="Green" title="E47">A55</font></i></a>, <a class="txt" href="aff_2.html#E42:39_1"><i><font color="Green" title="E52">A61</font></i></a>, <a class="ref" href="aff_2.html#D1" target="_self" title="AFF_2:def.1">Def1</a></i>;<br/>
<a NAME="E45:39_1"/><b>then </b>
<font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="aff_2.html#E40:39_1"><i><font color="Green" title="E50">A59</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E46:39_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</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="E47:39_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c20">r</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="E48:39_1"/><b>then </b>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c20">r</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="E49:39_1"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c20">r</font>,<font color="Maroon" title="c8">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="E50:39_1"/><b>then </b>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E41:39_1"><i><font color="Green" title="E51">A60</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E51:39_1"/><b>then </b>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="aff_2.html#E44:39_1"><i><font color="Green" title="E53">A63</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E52:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_2.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="aff_2.html#E28:39_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="aff_2.html#E29:39_1"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


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


</div>
