<?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>set </b><font color="Maroon" title="c2">AS</font> =  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font>;<br/>
<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E22">A2</font></i>: 
 <a href="conmetr.html#NR8" title="CONMETR:NR.8">des_holds_in</a> <font color="Maroon" title="c1">MS</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">C</font> be   <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>;<br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:22_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E2:22_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E3:22_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E4:22_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font> )
 <b>and </b><br/><a NAME="E5:22_1_1"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font> )
 ;<br/><a NAME="E6:22_1_1"/><i><font color="Green" title="E28">A8</font></i>: 
( not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E4:22_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><div><i><font color="Green" title="E29">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ;<br/><a NAME="E2:22_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:22_1_1"><i><font color="Green" title="E27">A7</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="E3:22_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c6">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/><a NAME="E4:22_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E3:22_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="euclmetr.html#E6:22_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E5:22_1_1_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><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="22_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1_2"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c12">aa</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c14">bb</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c16">cc</font> = <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c17">c1</font> = <font color="Maroon" title="c11">c'</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/><a NAME="E3:22_1_1_2"/><i><font color="Green" title="E31">A11</font></i>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c14">bb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_2_1"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c14">bb</font>
 ;<br/>

<a NAME="E2:22_1_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:22_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E3:22_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="euclmetr.html#E1:22_1_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E4:22_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:22_1_1_2"/><i><font color="Green" title="E32">A12</font></i>: 
not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">cc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_2_2"/>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c16">cc</font>
 ;<br/>

<a NAME="E2:22_1_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:22_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E3:22_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="euclmetr.html#E1:22_1_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="euclmetr.html#E4:22_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:22_1_1_2"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">cc</font>,<font color="Maroon" title="c17">c1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1_2_3"/>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c10">b'</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">c'</font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E1:22_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="euclmetr.html#E2:22_1_1"><i><font color="Green" title="E24">A4</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="E2:22_1_1_2_3"/>
( <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c13">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">cc</font>,<font color="Maroon" title="c17">c1</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><a NAME="E6:22_1_1_2"/>
( <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c14">bb</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c12">aa</font>,<font color="Maroon" title="c16">cc</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c17">c1</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:22_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/><a NAME="E7:22_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c16">cc</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">c1</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="euclmetr.html#E3:22_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="euclmetr.html#E4:22_1_1_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="euclmetr.html#E5:22_1_1_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="conmetr.html#D8" title="CONMETR:def.8">CONMETR:def 8</a></i>;<br/><b>hence </b><a NAME="E8:22_1_1_2"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 <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="E9:22_1_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E7:22_1_1"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a> 
 <b>by </b><i><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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E22">A14</font></i>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">c1</font> be    <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 <font color="Maroon" title="c1">MS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:22_2_1"/><i><font color="Green" title="E23">A15</font></i>: 
( not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">b</font> &amp; not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c7">c</font> )
 <b>and </b><br/><a NAME="E2:22_2_1"/><i><font color="Green" title="E24">A16</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">c1</font> )
 <b>and </b><br/><a NAME="E3:22_2_1"/><i><font color="Green" title="E25">A17</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c8">c1</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c10">a1'</font> = <font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c12">b1'</font> = <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c13">c'</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c14">c1'</font> = <font color="Maroon" title="c8">c1</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="E5:22_2_1"/><i><font color="Green" title="E26">A18</font></i>: 
( not <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> &amp; not <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c13">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_1_1"/>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> or <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c13">c'</font> )
 ;<br/>

<a NAME="E2:22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_2_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:22_2_1"/><i><font color="Green" title="E27">A19</font></i>: 
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">b1'</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">c1'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:22_2_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/><a NAME="E7:22_2_1"/><i><font color="Green" title="E28">A20</font></i>: 
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a1'</font>,<font color="Maroon" title="c12">b1'</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c13">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">a1'</font>,<font color="Maroon" title="c14">c1'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:22_2_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/><a NAME="E8:22_2_1"/><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c9">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a1'</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:22_2_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/><b>set </b><font color="Maroon" title="c15">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font>;<br/><a NAME="E9:22_2_1"/><i><font color="Green" title="E30">A22</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> &amp; <font color="Maroon" title="c10">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E8:22_2_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E10:22_2_1"/><b>then </b><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">P</font> being   <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> such that </b><br/><a NAME="E12:22_2_1"/><i><font color="Green" title="E32">A24</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font>
 <b>and </b><br/><a NAME="E13:22_2_1"/><i><font color="Green" title="E33">A25</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">P</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E9:22_2_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">C</font> being   <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> such that </b><br/><a NAME="E15:22_2_1"/><i><font color="Green" title="E34">A26</font></i>: 
<font color="Maroon" title="c13">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C</font>
 <b>and </b><br/><a NAME="E16:22_2_1"/><i><font color="Green" title="E35">A27</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c17">C</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E9:22_2_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><a NAME="E17:22_2_1"/><i><font color="Green" title="E36">A28</font></i>: 
( <font color="Maroon" title="c16">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c17">C</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#E13:22_2_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="euclmetr.html#E16:22_2_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E18:22_2_1"/>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">C</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E10:22_2_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="euclmetr.html#E13:22_2_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="euclmetr.html#E16:22_2_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><a NAME="E19:22_2_1"/><b>then </b>
( <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c12">b1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">c1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c17">C</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E6:22_2_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="euclmetr.html#E8:22_2_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><a NAME="E20:22_2_1"/><b>then </b><i><font color="Green" title="E37">A29</font></i>: 
( <font color="Maroon" title="c12">b1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">P</font> &amp; <font color="Maroon" title="c14">c1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C</font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E12:22_2_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="euclmetr.html#E15:22_2_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="euclmetr.html#E17:22_2_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E21:22_2_1"/><i><font color="Green" title="E38">A30</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">P</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:22_2_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="euclmetr.html#E9:22_2_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="euclmetr.html#E12:22_2_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/><a NAME="E22:22_2_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">a1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">C</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:22_2_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="euclmetr.html#E9:22_2_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="euclmetr.html#E15:22_2_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="aff_1.html#T65" title="AFF_1:th.65">AFF_1:65</a></i>;<br/><a NAME="E23:22_2_1"/><b>then </b>
<font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c13">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">b1'</font>,<font color="Maroon" title="c14">c1'</font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="euclmetr.html#E7:22_2_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="euclmetr.html#E9:22_2_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="euclmetr.html#E12:22_2_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="euclmetr.html#E13:22_2_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="euclmetr.html#E15:22_2_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="euclmetr.html#E16:22_2_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="euclmetr.html#E17:22_2_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="txt" href="euclmetr.html#E20:22_2_1"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="euclmetr.html#E21:22_2_1"><i><font color="Green" title="E38">A30</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/><b>hence </b><a NAME="E24:22_2_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c8">c1</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="E3:22_2"/>
 <a href="conmetr.html#NR8" title="CONMETR:NR.8">des_holds_in</a> <font color="Maroon" title="c1">MS</font>
 <b>by </b><i><a class="ref" href="conmetr.html#D8" title="CONMETR:def.8">CONMETR:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
(  <a href="conmetr.html#NR8" title="CONMETR:NR.8">des_holds_in</a> <font color="Maroon" title="c1">MS</font> iff  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">MS</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a>  )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:22"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>


</div>
