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

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

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="conmetr.html#NR4" title="CONMETR:NR.4">MH2_holds_in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="conmetr.html#D7" target="_self" title="CONMETR:def.7">CONMETR:def 7</a><br/><b>let </b><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="E2:12"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c10">M</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</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="E3:12"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c10">M</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  &amp; <font color="Maroon" title="c11">N</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  )
 
<b>by </b><i><a class="ref" href="analmetr.html#T62" title="ANALMETR:th.62">ANALMETR:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">b4'</font> = <font color="Maroon" title="c9">b4</font>, <font color="Maroon" title="c13">b1'</font> = <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c14">b2'</font> = <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c15">b3'</font> = <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c16">a1'</font> = <font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c17">a2'</font> = <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c18">a3'</font> = <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c19">a4'</font> = <font color="Maroon" title="c5">a4</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">M'</font> = <font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c21">N'</font> = <font color="Maroon" title="c11">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c20">M'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c21">N'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 
<b>by </b><i><a class="txt" href="conmetr.html#E3:12"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></i>;<br/>
<a NAME="E7:12"/>
not <font color="Maroon" title="c20">M'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c20">M'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N'</font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c10">M</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a> <font color="Maroon" title="c11">N</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64" title="ANALMETR:th.64">ANALMETR:64</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T73" title="ANALMETR:th.73">ANALMETR:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:12"/><b>then </b>
 ex <font color="Olive" title="b1">o'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">o'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">M'</font> &amp; <font color="Olive" title="b1">o'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N'</font> )
 
<b>by </b><i><a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">o</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="E10:12"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c23">o'</font> = <font color="Maroon" title="c22">o</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<div><i><font color="Green" title="E20">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b4</font>
 ;<br/><a NAME="E2:12_2"/><i><font color="Green" title="E22">A8</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c17">a2'</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E10:12"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E3:12_2"/><i><font color="Green" title="E23">A9</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c19">a4'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_2_1"/>
<font color="Maroon" title="c21">N'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c21">N'</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/>
<a NAME="E2:12_2_1"/><b>then </b>
<font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c17">a2'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c19">a4'</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E10:12"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c19">a4'</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="E4:12_2"/>
<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c17">a2'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c19">a4'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_2_2"/>
<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="c7">b2</font>,<font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E1:12_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E2:12_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="c2">a1</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E3:12_2_2"/><b>then </b>
<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="c5">a4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_2"/>
<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c17">a2'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c19">a4'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_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="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E1:12_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="conmetr.html#E2:12_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="conmetr.html#E3:12_2"><i><font color="Green" title="E23">A9</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b4</font>
 ;<br/><div><i><font color="Green" title="E22">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_1"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b3</font>
 ;<br/><a NAME="E2:12_3_1"/><i><font color="Green" title="E24">A13</font></i>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c16">a1'</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E10:12"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><a NAME="E3:12_3_1"/><i><font color="Green" title="E25">A14</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c18">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_1_1"/>
<font color="Maroon" title="c20">M'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c20">M'</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/>
<a NAME="E2:12_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c16">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c18">a3'</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E10:12"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c23">o'</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c18">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/>


</div><b>end;</b></div><a NAME="E4:12_3_1"/>
<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c16">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c18">a3'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_1_2"/>
<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="c8">b3</font>,<font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E1:12_3_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E2:12_3_1_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="c4">a3</font>,<font color="Maroon" title="c3">a2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E3:12_3_1_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="c4">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E4:12_3_1_2"/>
<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c16">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c18">a3'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_3_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="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E1:12_3_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="conmetr.html#E3:12_3_1"><i><font color="Green" title="E25">A14</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_3_2"/><i><font color="Green" title="E23">A15</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="E2:12_3_2"/><i><font color="Green" title="E24">A16</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_2_1"/>
( <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/>

<a NAME="E2:12_3_2_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="c8">b3</font>,<font color="Maroon" title="c7">b2</font> or <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="c7">b2</font>,<font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:12_3_2_1"/><b>then </b>
( <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</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> or <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="c6">b1</font>,<font color="Maroon" title="c9">b4</font> )
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_1"/><b>then </b>
( <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</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> or <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="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:12_3_2_1"/><b>then </b>
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c6">b1</font> or  <a href="analmetr.html#R5" title="ANALMETR:pred.5">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="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E6:12_3_2_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">b2'</font>,<font color="Maroon" title="c15">b3'</font>,<font color="Maroon" title="c13">b1'</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">b1'</font>,<font color="Maroon" title="c14">b2'</font>,<font color="Maroon" title="c12">b4'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E7:12_3_2_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">b1'</font>,<font color="Maroon" title="c15">b3'</font>,<font color="Maroon" title="c14">b2'</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">b2'</font>,<font color="Maroon" title="c12">b4'</font>,<font color="Maroon" title="c13">b1'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E8:12_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E1:12_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="conmetr.html#E1:12_3_2"><i><font color="Green" title="E23">A15</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>consider </b><font color="Maroon" title="c24">d3'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E4:12_3_2"/><i><font color="Green" title="E25">A17</font></i>: 
( <font color="Maroon" title="c23">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">d3'</font> &amp; <font color="Maroon" title="c24">d3'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N'</font> )
 <b>by </b><i><a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</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="c25">d3</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="E6:12_3_2"/><i><font color="Green" title="E26">A18</font></i>: 
( <font color="Maroon" title="c25">d3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c25">d3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">o</font> )
 <b>by </b><i><a class="txt" href="conmetr.html#E4:12_3_2"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">d3'</font> = <font color="Maroon" title="c25">d3</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><b>consider </b><font color="Maroon" title="c27">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="E9:12_3_2"/><i><font color="Green" title="E27">A19</font></i>: 
( <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c27">e1</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c27">e1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">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="E11:12_3_2"/><i><font color="Green" title="E28">A20</font></i>: 
( <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c28">e2</font> &amp; <font color="Maroon" title="c25">d3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">e2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">e1'</font> = <font color="Maroon" title="c27">e1</font>, <font color="Maroon" title="c30">e2'</font> = <font color="Maroon" title="c28">e2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E13:12_3_2"/>
not <font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c29">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c26">d3'</font>,<font color="Maroon" title="c30">e2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_2_2"/>
<font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c29">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c26">d3'</font>,<font color="Maroon" title="c30">e2'</font>
 ;<br/>

<a NAME="E2:12_3_2_2"/><b>then </b>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c27">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c28">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:12_3_2_2"/><b>then </b>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c28">e2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E9:12_3_2"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_2"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E11:12_3_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>
<a NAME="E5:12_3_2_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T78" title="ANALMETR:th.78">ANALMETR:78</a></i>;<br/>
<a NAME="E6:12_3_2_2"/><b>then </b>
<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="c3">a2</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="conmetr.html#E4:12_3_2_2"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/>
<a NAME="E7:12_3_2_2"/><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="c4">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E8:12_3_2_2"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c4">a3</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E9:12_3_2_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c19">a4'</font>,<font color="Maroon" title="c18">a3'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E10:12_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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="c31">d2'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E15:12_3_2"/><i><font color="Green" title="E29">A22</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c29">e1'</font>,<font color="Maroon" title="c31">d2'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c26">d3'</font>,<font color="Maroon" title="c30">e2'</font>,<font color="Maroon" title="c31">d2'</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>reconsider </b><font color="Maroon" title="c32">d2</font> = <font color="Maroon" title="c31">d2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E17:12_3_2"/><i><font color="Green" title="E30">A23</font></i>: 
<font color="Maroon" title="c32">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_3"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c27">e1</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E15:12_3_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_3"/><b>then </b>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c27">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:12_3_2_3"/><b>then </b>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E9:12_3_2"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_3"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E5:12_3_2_3"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c31">d2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E6:12_3_2_3"/>
<font color="Maroon" title="c32">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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><a NAME="E18:12_3_2"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c32">d2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_4"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c28">e2</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E15:12_3_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_4"/><b>then </b>
<font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c28">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c32">d2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_2_4"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c32">d2</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E11:12_3_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c33">d2</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="E20:12_3_2"/><i><font color="Green" title="E31">A24</font></i>: 
( <font color="Maroon" title="c33">d2</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>,<font color="Maroon" title="c3">a2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c33">d2</font> )
 <b>by </b><i><a class="txt" href="conmetr.html#E17:12_3_2"><i><font color="Green" title="E30">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">d2'</font> = <font color="Maroon" title="c33">d2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><b>consider </b><font color="Maroon" title="c35">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="E23:12_3_2"/><i><font color="Green" title="E32">A25</font></i>: 
( <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="c35">e1</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="c35">e1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></i>;<br/><b>consider </b><font color="Maroon" title="c36">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="E25:12_3_2"/><i><font color="Green" title="E33">A26</font></i>: 
( <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c36">e2</font> &amp; <font color="Maroon" title="c33">d2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">e2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c37">e1'</font> = <font color="Maroon" title="c35">e1</font>, <font color="Maroon" title="c38">e2'</font> = <font color="Maroon" title="c36">e2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E27:12_3_2"/>
not <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c37">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c34">d2'</font>,<font color="Maroon" title="c38">e2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_2_5"/>
<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c37">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c34">d2'</font>,<font color="Maroon" title="c38">e2'</font>
 ;<br/>

<a NAME="E2:12_3_2_5"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c35">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c36">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:12_3_2_5"/><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="c33">d2</font>,<font color="Maroon" title="c36">e2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E23:12_3_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_5"/><b>then </b><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E25:12_3_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>
<a NAME="E5:12_3_2_5"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T78" title="ANALMETR:th.78">ANALMETR:78</a></i>;<br/>
<a NAME="E6:12_3_2_5"/><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="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="conmetr.html#E4:12_3_2_5"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/>
<a NAME="E7:12_3_2_5"/><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="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E8:12_3_2_5"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">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="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E9:12_3_2_5"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c17">a2'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E10:12_3_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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="c39">d1'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E29:12_3_2"/><i><font color="Green" title="E34">A28</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c37">e1'</font>,<font color="Maroon" title="c39">d1'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c34">d2'</font>,<font color="Maroon" title="c38">e2'</font>,<font color="Maroon" title="c39">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>reconsider </b><font color="Maroon" title="c40">d1</font> = <font color="Maroon" title="c39">d1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E31:12_3_2"/><i><font color="Green" title="E35">A29</font></i>: 
<font color="Maroon" title="c40">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_6"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c35">e1</font>,<font color="Maroon" title="c40">d1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E29:12_3_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_6"/><b>then </b>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c35">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c40">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:12_3_2_6"/><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="c40">d1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E23:12_3_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_6"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c40">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E5:12_3_2_6"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c19">a4'</font>,<font color="Maroon" title="c39">d1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E6:12_3_2_6"/>
<font color="Maroon" title="c40">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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><a NAME="E32:12_3_2"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c40">d1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_7"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c36">e2</font>,<font color="Maroon" title="c40">d1</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E29:12_3_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_7"/><b>then </b>
<font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c36">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c40">d1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_2_7"/>
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c40">d1</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E25:12_3_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c41">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="E34:12_3_2"/><i><font color="Green" title="E37">A31</font></i>: 
( <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="c41">e1</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="c41">e1</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></i>;<br/><b>consider </b><font color="Maroon" title="c42">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="E36:12_3_2"/><i><font color="Green" title="E38">A32</font></i>: 
( <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c42">e2</font> &amp; <font color="Maroon" title="c40">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c42">e2</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c43">e1'</font> = <font color="Maroon" title="c41">e1</font>, <font color="Maroon" title="c44">e2'</font> = <font color="Maroon" title="c42">e2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E38:12_3_2"/>
not <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c43">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c39">d1'</font>,<font color="Maroon" title="c44">e2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_2_8"/>
<font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c43">e1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c39">d1'</font>,<font color="Maroon" title="c44">e2'</font>
 ;<br/>

<a NAME="E2:12_3_2_8"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c41">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c42">e2</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E3:12_3_2_8"/><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="c40">d1</font>,<font color="Maroon" title="c42">e2</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E34:12_3_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_8"/><b>then </b><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E36:12_3_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>
<a NAME="E5:12_3_2_8"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T78" title="ANALMETR:th.78">ANALMETR:78</a></i>;<br/>
<a NAME="E6:12_3_2_8"/><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="c2">a1</font>,<font color="Maroon" title="c5">a4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="conmetr.html#E4:12_3_2_8"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/>
<a NAME="E7:12_3_2_8"/><b>then </b>
<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c2">a1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E8:12_3_2_8"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E9:12_3_2_8"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c19">a4'</font>,<font color="Maroon" title="c17">a2'</font>,<font color="Maroon" title="c16">a1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E10:12_3_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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="c45">d4'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E40:12_3_2"/><i><font color="Green" title="E39">A34</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c43">e1'</font>,<font color="Maroon" title="c45">d4'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c39">d1'</font>,<font color="Maroon" title="c44">e2'</font>,<font color="Maroon" title="c45">d4'</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>reconsider </b><font color="Maroon" title="c46">d4</font> = <font color="Maroon" title="c45">d4'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/><a NAME="E42:12_3_2"/><i><font color="Green" title="E40">A35</font></i>: 
<font color="Maroon" title="c46">d4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_9"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c41">e1</font>,<font color="Maroon" title="c46">d4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E40:12_3_2"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_9"/><b>then </b>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c41">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c46">d4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E3:12_3_2_9"/><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="c46">d4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E34:12_3_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="analmetr.html#T82" title="ANALMETR:th.82">ANALMETR:82</a></i>;<br/>
<a NAME="E4:12_3_2_9"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c46">d4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E5:12_3_2_9"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c16">a1'</font>,<font color="Maroon" title="c18">a3'</font>,<font color="Maroon" title="c45">d4'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E6:12_3_2_9"/>
<font color="Maroon" title="c46">d4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E2:12_3_2"><i><font color="Green" title="E24">A16</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><a NAME="E43:12_3_2"/><i><font color="Green" title="E41">A36</font></i>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c46">d4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2_10"><b>proof </b></a><div class="add">

<a NAME="E1:12_3_2_10"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c42">e2</font>,<font color="Maroon" title="c46">d4</font>
 
<b>by </b><i><a class="txt" href="conmetr.html#E40:12_3_2"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E2:12_3_2_10"/><b>then </b>
<font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c42">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c46">d4</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_2_10"/>
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c46">d4</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E36:12_3_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/>


</div><b>end;</b></div><a NAME="E44:12_3_2"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c46">d4</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E1:12"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12_3_2"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="conmetr.html#E20:12_3_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="conmetr.html#E32:12_3_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="conmetr.html#E31:12_3_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="conmetr.html#E43:12_3_2"><i><font color="Green" title="E41">A36</font></i></a>, <a class="txt" href="conmetr.html#E42:12_3_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="ref" href="conmetr.html#D4" target="_self" title="CONMETR:def.4">Def4</a></i>;<br/><a NAME="E45:12_3_2"/><i><font color="Green" title="E43">A38</font></i>: 
<font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c33">d2</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E20:12_3_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/><a NAME="E46:12_3_2"/><i><font color="Green" title="E44">A39</font></i>: 
<font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c33">d2</font>,<font color="Maroon" title="c40">d1</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E32:12_3_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/><a NAME="E47:12_3_2"/>
<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c40">d1</font>,<font color="Maroon" title="c46">d4</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E43:12_3_2"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="analmetr.html#T84" title="ANALMETR:th.84">ANALMETR:84</a></i>;<br/><a NAME="E48:12_3_2"/><b>then </b><i><font color="Green" title="E45">A40</font></i>: 
<font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c25">d3</font>,<font color="Maroon" title="c46">d4</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E1:12"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12_3_2"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="conmetr.html#E20:12_3_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="conmetr.html#E31:12_3_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="conmetr.html#E42:12_3_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="conmetr.html#E45:12_3_2"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="conmetr.html#E46:12_3_2"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="conmetr.html#D4" target="_self" title="CONMETR:def.4">Def4</a></i>;<br/><a NAME="E49:12_3_2"/>
<font color="Maroon" title="c25">d3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c46">d4</font>
 <b>by </b><i><a class="txt" href="conmetr.html#E2:12"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conmetr.html#E6:12"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E10:12"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="conmetr.html#E6:12_3_2"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="conmetr.html#E42:12_3_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><b>hence </b><a NAME="E50:12_3_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="conmetr.html#E44:12_3_2"><i><font color="Green" title="E42">A37</font></i></a>, <a class="txt" href="conmetr.html#E48:12_3_2"><i><font color="Green" title="E45">A40</font></i></a>, <a class="ref" href="analmetr.html#T85" title="ANALMETR:th.85">ANALMETR:85</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:12_3"/>
<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="conmetr.html#E2:12_3"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:12"/>
<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="conmetr.html#E12:12"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>


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