<?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="E1:19"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">AP</font> <a href="aff_2.html#NR4" title="AFF_2:NR.4">satisfies_DES</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_3.html#D1" target="_self" title="AFF_3:def.1">AFF_3:def 1</a><br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>;<br/><b>let </b><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">c'</font>, <font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font>;<br/>























<b>assume </b><b>that </b><br/><a NAME="E2:19"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> )
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="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="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">q</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a'</font> )
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green" title="E13">A5</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c12">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c14">K</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font>;<br/>
<b>set </b><font color="Maroon" title="c15">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E14">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="c8">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">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; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="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">b'</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E16">A8</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:19"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:19"/><i><font color="Green" title="E17">A9</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="c7">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font>
 ;<br/>

<a NAME="E2:19_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:19_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E8:19"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="aff_3.html#E1:19_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:19"/><i><font color="Green" title="E18">A11</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="c11">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E19">A12</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E2:19_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E8:19"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:19_3"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E9:19"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E1:19_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="aff_1.html#T69" title="AFF_1:th.69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:19"/><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_4"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font>
 ;<br/>

<a NAME="E2:19_4"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">a'</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c7">a'</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">o</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E8:19"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:19_4"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:19"/><i><font color="Green" title="E20">A14</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:19"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E7:19"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E9:19"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="aff_3.html#E10:19"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">D</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font>;<br/>
<a NAME="E13:19"/><i><font color="Green" title="E21">A15</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">D'</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="E15:19"/><i><font color="Green" title="E22">A16</font></i>: 
( <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D'</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">D'</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="E16:19"/><i><font color="Green" title="E23">A17</font></i>: 
( <font color="Maroon" title="c17">D'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c17">D'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E17:19"/>
not <font color="Maroon" title="c17">D'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_5"/>
<font color="Maroon" title="c17">D'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E2:19_5"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<a NAME="E3:19_5"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E13:19"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:19_5"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:19"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c18">d</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="E19:19"/><i><font color="Green" title="E24">A18</font></i>: 
( <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D'</font> &amp; <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E16:19"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<a NAME="E20:19"/><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_6"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b'</font>
 ;<br/>

<a NAME="E2:19_6"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">D'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E7:19"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="aff_3.html#E16:19"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="aff_3.html#E19:19"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>
<a NAME="E3:19_6"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c11">c'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:19"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E13:19"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E4:19_6"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E7:19"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E13:19"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E5:19_6"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E11:19"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E6:19_6"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:19"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:19"/>
( <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c18">d</font> &amp; <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E13:19"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="aff_3.html#E19:19"><i><font color="Green" title="E24">A18</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#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E22:19"/><b>then </b><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c7">a'</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E1:19"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E3:19"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E19:19"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></i>;<br/>
<a NAME="E23:19"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</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="E24:19"/><b>then </b>
( <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c7">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c13">q</font> )
 
<b>by </b><i><a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E13:19"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="aff_3.html#E15:19"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="aff_3.html#E19:19"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="aff_3.html#E22:19"><i><font color="Green" title="E26">A20</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#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E25:19"/><b>then </b>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c11">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="aff_3.html#E1:19"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="aff_3.html#E2:19"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="aff_3.html#E4:19"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="aff_3.html#E7:19"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="aff_3.html#E12:19"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="aff_3.html#E19:19"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="aff_3.html#E20:19"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="aff_2.html#D4" title="AFF_2:def.4">AFF_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E26:19"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c10">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="aff_3.html#E5:19"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="aff_3.html#E6:19"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>


</div>
