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

<b>let </b><font color="Maroon" title="c1">MS</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E24">A1</font></i>: 
( not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c1</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">X</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Y</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font>;<br/>
<a NAME="E2:27"/>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="euclmetr.html#T1" target="_self" title="EUCLMETR:th.1">Th1</a></i>;<br/>
<a NAME="E3:27"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
(  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  )
 
<b>by </b><i><a class="ref" href="analmetr.html#D13" title="ANALMETR:def.13">ANALMETR:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">X'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c8">X'</font> )
 <b>by </b><i><a class="ref" href="conmetr.html#T3" title="CONMETR:th.3">CONMETR:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Y'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y'</font> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c9">Y'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="conmetr.html#T3" title="CONMETR:th.3">CONMETR:3</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c8">X'</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  &amp; <font color="Maroon" title="c9">Y'</font> <a href="analmetr.html#NR6" title="ANALMETR:NR.6">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="euclmetr.html#E7:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="analmetr.html#T62" title="ANALMETR:th.62">ANALMETR:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">o'</font> = <font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c5">a</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">MS</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="E10:27"/>
(  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c12">a'</font> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">o'</font>,<font color="Maroon" title="c12">a'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T56" title="ANALMETR:th.56">ANALMETR:56</a></i>;<br/>
<a NAME="E11:27"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E30">A7</font></i>: 
not <font color="Maroon" title="c8">X'</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a> <font color="Maroon" title="c9">Y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c8">X'</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a> <font color="Maroon" title="c9">Y'</font>
 ;<br/>

<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c8">X'</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E7:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="analmetr.html#T73" title="ANALMETR:th.73">ANALMETR:73</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
 <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="analmetr.html#T87" title="ANALMETR:th.87">ANALMETR:87</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">X1</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c14">Y1</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</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="E5:27_1"/>
<font color="Maroon" title="c13">X1</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">Y1</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E3:27_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="analmetr.html#T64" title="ANALMETR:th.64">ANALMETR:64</a></i>;<br/>
<a NAME="E6:27_1"/><b>then </b>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E11:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E7:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="euclmetr.html#E3:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="conmetr.html#T4" title="CONMETR:th.4">CONMETR:4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c13">X1</font> = <font color="Maroon" title="c8">X'</font>, <font color="Maroon" title="c14">Y1</font> = <font color="Maroon" title="c9">Y'</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">MS</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="E14:27"/><i><font color="Green" title="E31">A9</font></i>: 
( <font color="Maroon" title="c13">X1</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c14">Y1</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E8:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></i>;<br/>
<a NAME="E15:27"/>
not <font color="Maroon" title="c13">X1</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c14">Y1</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E12:27"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="analmetr.html#T64" title="ANALMETR:th.64">ANALMETR:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">a1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font></span>)</span><b> such that </b><br/><a NAME="E17:27"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c15">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> &amp; <font color="Maroon" title="c15">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E14:27"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a1</font> = <font color="Maroon" title="c15">a1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">a1</font>
;<br/>

<b>thus </b><a NAME="E19:27"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c16">a1</font> &amp; <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c16">a1</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="euclmetr.html#E7:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="euclmetr.html#E11:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="euclmetr.html#E17:27"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="analmetr.html#T78" title="ANALMETR:th.78">ANALMETR:78</a></i>;<br/>


</div>
