<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="aff_2.html#NR11" title="AFF_2:NR.11">satisfies_des</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">b4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> )
 ;<br/><a NAME="E2:18_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c10">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c11">N</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E3:18_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c11">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><div><i><font color="Green" title="E16">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a3</font> or <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a4</font> )
 ;<br/><div><i><font color="Green" title="E18">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a3</font>
 ;<br/><a NAME="E2:18_1_1_1"/>
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1_1_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b3</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1"/><i><font color="Green" title="E21">A10</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c8">b3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_1_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E2:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</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="E3:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E5:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E6:18_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</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="E7:18_1_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c8">b3</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><b>end;</b></div>
<a NAME="E3:18_1_1_1_1"/>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c8">b3</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">N</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_1_1_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_1_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_1_1"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a4</font>
 ;<br/><a NAME="E2:18_1_1_2"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c9">b4</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><a NAME="E3:18_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/><a NAME="E4:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</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="E5:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E6:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</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="E7:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E8:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</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="E9:18_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E10:18_1_1_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green" title="E17">A13</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a3</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a4</font> )
 ;<br/><div><i><font color="Green" title="E18">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green" title="E19">A15</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> or  not <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> )
 ;<br/><a NAME="E2:18_1_2_1"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1"/><b>then </b><i><font color="Green" title="E20">A16</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c6">b1</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><i><font color="Green" title="E21">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_1"/><i><font color="Green" title="E22">A18</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font>
 ;<br/><a NAME="E2:18_1_2_1_1"/><b>then </b><i><font color="Green" title="E23">A19</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E3:18_1_2_1_1"/>
<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c9">b4</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><a NAME="E4:18_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A20</font></i>: 
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_1_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/><a NAME="E5:18_1_2_1_1"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E6:18_1_2_1_1"/><b>then </b><i><font color="Green" title="E25">A21</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">b2</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="E7:18_1_2_1_1"/>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">b2</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><a NAME="E8:18_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_1_1"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/><a NAME="E9:18_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E10:18_1_2_1_1"/><b>then </b><i><font color="Green" title="E26">A22</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c8">b3</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="E11:18_1_2_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c8">b3</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">N</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><a NAME="E12:18_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E10:18_1_2_1_1"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/><b>hence </b><a NAME="E13:18_1_2_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E4:18_1_2_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font>
 ;<br/><a NAME="E2:18_1_2_1_2"/>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1_2"/><b>then </b><i><font color="Green" title="E23">A24</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c7">b2</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><i><font color="Green" title="E24">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2_1"/><i><font color="Green" title="E25">A26</font></i>: 
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font>
 ;<br/><a NAME="E2:18_1_2_1_2_1"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E26">A27</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E4:18_1_2_1_2_1"/>
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_1_2_1_1"/>
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 ;<br/>

<a NAME="E2:18_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E27">A28</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:18_1_2_1_2_1_1"/>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c7">b2</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">M</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1_2_1"><i><font color="Green" title="E25">A26</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_1_2_1_1"><i><font color="Green" title="E27">A28</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:18_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E27">A29</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1_2_1_2_1"><i><font color="Green" title="E26">A27</font></i></a></i>;<br/><a NAME="E6:18_1_2_1_2_1"/>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E7:18_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E28">A30</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c9">b4</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="E8:18_1_2_1_2_1"/>
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_1_2_1_2"/>
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b4</font>
 ;<br/>

<a NAME="E2:18_1_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E3:18_1_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E29">A31</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:18_1_2_1_2_1_2"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c6">b1</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">N</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1_2_1_2_1_2"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:18_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E29">A32</font></i>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="conmetr1.html#E7:18_1_2_1_2_1"><i><font color="Green" title="E28">A30</font></i></a></i>;<br/><a NAME="E10:18_1_2_1_2_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_2_1_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1_2_1"><i><font color="Green" title="E25">A26</font></i></a>, <a class="ref" href="aff_1.html#T11" title="AFF_1:th.11">AFF_1:11</a></i>;<br/><a NAME="E11:18_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</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="E12:18_1_2_1_2_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E5:18_1_2_1_2_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="conmetr1.html#E9:18_1_2_1_2_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1_2_2"/>
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font>
 ;<br/><a NAME="E2:18_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E25">A33</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E3:18_1_2_1_2_2"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c6">b1</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">N</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_1_2_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E4:18_1_2_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_1_2_2"><i><font color="Green" title="E25">A33</font></i></a>, <a class="ref" href="aff_1.html#T60" title="AFF_1:th.60">AFF_1:60</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_2_1_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1_2_1_2"><i><font color="Green" title="E23">A24</font></i></a>, <a class="txt" href="conmetr1.html#E4:18_1_2_1_2"><i><font color="Green" title="E24">A25</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_2_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="conmetr1.html#E3:18_1_2_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="conmetr1.html#E4:18_1_2_1"><i><font color="Green" title="E21">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c12">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:18_1_2_2"/><i><font color="Green" title="E19">A34</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">d</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="translac.html#T2" title="TRANSLAC:th.2">TRANSLAC:2</a></i>;<br/><a NAME="E4:18_1_2_2"/>
 ex <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c13">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:18_1_2_2_1"/><i><font color="Green" title="E20">A35</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">Y</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">Y</font>
;<br/>

<b>thus </b><a NAME="E3:18_1_2_2_1"/>
( <font color="Maroon" title="c13">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E2:18_1_2_2_1"><i><font color="Green" title="E20">A35</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c13">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:18_1_2_2"/><i><font color="Green" title="E20">A36</font></i>: 
( <font color="Maroon" title="c13">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> )
 ;<br/><a NAME="E7:18_1_2_2"/><i><font color="Green" title="E21">A37</font></i>: 
( <font color="Maroon" title="c13">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c11">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E8:18_1_2_2"/>
 ex <font color="Olive" title="b1">d1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Olive" title="b1">d1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c14">e1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:18_1_2_2_2"/><i><font color="Green" title="E22">A38</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">e1</font> &amp; <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:18_1_2_2_2"/><i><font color="Green" title="E23">A39</font></i>: 
( <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">e2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">e2</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></i>;<br/>
<a NAME="E5:18_1_2_2_2"/>
not <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_2_2_1"/>
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">e2</font>
 ;<br/>

<a NAME="E2:18_1_2_2_2_1"/><b>then </b><i><font color="Green" title="E24">A40</font></i>: 
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:18_1_2_2_2"><i><font color="Green" title="E23">A39</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E3:18_1_2_2_2_1"/>
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">e1</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">Y</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E7:18_1_2_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_2"><i><font color="Green" title="E22">A38</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E4:18_1_2_2_2_1"/><b>then </b><i><font color="Green" title="E25">A41</font></i>: 
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">Y</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E2:18_1_2_2_2"><i><font color="Green" title="E22">A38</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_2_1"><i><font color="Green" title="E24">A40</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/>
<a NAME="E5:18_1_2_2_2_1"/>
<font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">Y</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a></i>;<br/>
<a NAME="E6:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c13">Y</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/>
<a NAME="E7:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E4:18_1_2_2_2_1"><i><font color="Green" title="E25">A41</font></i></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></i>;<br/>
<a NAME="E8:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</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="E9:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E10:18_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</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="E11:18_1_2_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:18_1_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c16">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:18_1_2_2_2"/><i><font color="Green" title="E24">A42</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">e1</font>,<font color="Maroon" title="c16">d1</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">e2</font>,<font color="Maroon" title="c16">d1</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74" title="AFF_1:th.74">AFF_1:74</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">d1</font>
;<br/>

<a NAME="E8:18_1_2_2_2"/>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c15">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c16">d1</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E7:18_1_2_2_2"><i><font color="Green" title="E24">A42</font></i></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="E9:18_1_2_2_2"/>
( <font color="Maroon" title="c16">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c16">d1</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_2"><i><font color="Green" title="E22">A38</font></i></a>, <a class="txt" href="conmetr1.html#E4:18_1_2_2_2"><i><font color="Green" title="E23">A39</font></i></a>, <a class="txt" href="conmetr1.html#E7:18_1_2_2_2"><i><font color="Green" title="E24">A42</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#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E10:18_1_2_2"/><i><font color="Green" title="E22">A43</font></i>: 
( <font color="Maroon" title="c14">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c14">d1</font> )
 ;<br/><a NAME="E11:18_1_2_2"/><i><font color="Green" title="E23">A44</font></i>: 
( <font color="Maroon" title="c11">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c11">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">M</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E12:18_1_2_2"/><i><font color="Green" title="E24">A45</font></i>: 
( <font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2_2_3"/><i><font color="Green" title="E25">A46</font></i>: 
(  not <font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M</font> or  not <font color="Maroon" title="c11">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Y</font> or  not <font color="Maroon" title="c13">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M</font> )
 ;<br/>

<div><i><font color="Green" title="E26">A47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2_3_1"/>
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</font>
 ;<br/><a NAME="E2:18_1_2_2_3_1"/><b>then </b><i><font color="Green" title="E27">A48</font></i>: 
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E7:18_1_2_2"><i><font color="Green" title="E21">A37</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_2_3_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E4:18_1_2_2_3_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</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="E5:18_1_2_2_3_1"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_3_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E6:18_1_2_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:18_1_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2_3_2"/>
<font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">M</font>
 ;<br/><a NAME="E2:18_1_2_2_3_2"/><b>then </b><i><font color="Green" title="E27">A49</font></i>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/><a NAME="E3:18_1_2_2_3_2"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</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="E4:18_1_2_2_3_2"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_3_2"><i><font color="Green" title="E27">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="E5:18_1_2_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:18_1_2_2_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_1_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1_2_2_3"><i><font color="Green" title="E25">A46</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1_2_2_3"><i><font color="Green" title="E26">A47</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_1_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</font></i></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E14:18_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</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="E15:18_1_2_2"/><b>then </b><i><font color="Green" title="E25">A50</font></i>: 
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/><a NAME="E16:18_1_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c14">d1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E10:18_1_2_2"><i><font color="Green" title="E22">A43</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="E17:18_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c14">d1</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="E18:18_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c14">d1</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="E19:18_1_2_2"/><b>then </b>
( <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c14">d1</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E15:18_1_2_2"><i><font color="Green" title="E25">A50</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/><a NAME="E20:18_1_2_2"/><b>then </b><i><font color="Green" title="E26">A51</font></i>: 
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c4">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c8">b3</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E10:18_1_2_2"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="conmetr1.html#E11:18_1_2_2"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="conmetr1.html#E12:18_1_2_2"><i><font color="Green" title="E24">A45</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/><a NAME="E21:18_1_2_2"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c14">d1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_2_2_4"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E3:18_1_2_2"><i><font color="Green" title="E19">A34</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="E2:18_1_2_2_4"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</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="E3:18_1_2_2_4"/><b>then </b>
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:18_1_2_2_4"/><b>then </b>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</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="E5:18_1_2_2_4"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c12">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c14">d1</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E10:18_1_2_2"><i><font color="Green" title="E22">A43</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="E22:18_1_2_2"/><b>then </b><i><font color="Green" title="E27">A52</font></i>: 
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">d1</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E10:18_1_2_2"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="conmetr1.html#E12:18_1_2_2"><i><font color="Green" title="E24">A45</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/><a NAME="E23:18_1_2_2"/>
<font color="Maroon" title="c13">Y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><b>hence </b><a NAME="E24:18_1_2_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="conmetr1.html#E1:18_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="conmetr1.html#E2:18_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="conmetr1.html#E6:18_1_2_2"><i><font color="Green" title="E20">A36</font></i></a>, <a class="txt" href="conmetr1.html#E10:18_1_2_2"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="conmetr1.html#E12:18_1_2_2"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="conmetr1.html#E20:18_1_2_2"><i><font color="Green" title="E26">A51</font></i></a>, <a class="txt" href="conmetr1.html#E22:18_1_2_2"><i><font color="Green" title="E27">A52</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E2:18_1_2"><i><font color="Green" title="E18">A14</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E4:18_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon" title="c1">X</font> <a href="conmetr1.html#NR1" title="CONMETR1:NR.1">satisfies_minor_SCH</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D1" target="_self" title="CONMETR1:def.1">Def1</a></i>;<br/>


</div>
