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

<b>let </b><font color="Maroon">X</font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<a NAME="E1:29"/><i><font color="Green">E45</font></i>: 
(  <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font> implies  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E46</font></i>: 
 <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">A</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">M</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">N</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font> )
 ;<br/><b>assume </b><a NAME="E2:29_1_1"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 ;<br/><b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o</font>, <font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a1</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b1</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">c1'</font> = <font color="Maroon">c1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E4:29_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E5:29_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a'</font>,<font color="Maroon">a1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c'</font>,<font color="Maroon">c1'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_1"/>
( <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:29_1_1_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a1</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b1</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_1"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a'</font>,<font color="Maroon">a1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c'</font>,<font color="Maroon">c1'</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:29_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2"/><i><font color="Green">E54</font></i>: 
(  not <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1'</font> or  not <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1'</font> or  not <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1'</font> )
 ;<br/>

<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 ;<br/><a NAME="E2:29_1_1_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 ;<br/>

<a NAME="E2:29_1_1_2_1_1"/><i><font color="Green">E59</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:29_1_1_2_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:29_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_2_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">o</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:29_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:29_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E6:29_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:29_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_2_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E9:29_1_1_2_1_1_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_1_2_1_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:29_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:29_1_1_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 ;<br/>

<a NAME="E2:29_1_1_2_1_2"/><i><font color="Green">E63</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_1_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:29_1_1_2_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:29_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_2_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">o</font>,<font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2_1_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:29_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:29_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:29_1_1_2_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:29_1_1_2_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E6:29_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E7:29_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_2_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E9:29_1_1_2_1_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_1_2_1_2_2"/>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:29_1_1_2_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:29_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font>
 ;<br/><a NAME="E2:29_1_1_2_2"/><i><font color="Green">E69</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:29_1_1_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:29_1_1_2_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:29_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_2_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:29_1_1_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E4:29_1_1_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E5:29_1_1_2_2"/>
<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:29_1_1_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:29_1_1_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_2_2_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E9:29_1_1_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:29_1_1_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E11:29_1_1_2_2_2"/>
<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:29_1_1_2_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1'</font>
 ;<br/><a NAME="E2:29_1_1_2_3"/><i><font color="Green">E73</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_2_3_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:29_1_1_2_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:29_1_1_2_3_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E4:29_1_1_2_3_1"/><b>then </b>
( <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_2_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_2_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:29_1_1_2_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E4:29_1_1_2_3"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E5:29_1_1_2_3"/>
<font color="Maroon">c</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2_3_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<a NAME="E2:29_1_1_2_3_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:29_1_1_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:29_1_1_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E5:29_1_1_2_3_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E6:29_1_1_2_3_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:29_1_1_2_3_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_2_3_2"/><b>then </b>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E9:29_1_1_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E10:29_1_1_2_3_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c</font>,<font color="Maroon">o</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E11:29_1_1_2_3_2"/>
<font color="Maroon">c</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:29_1_1_2_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E7:29_1_1"/><i><font color="Green">E76</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a'</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3"/>
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a'</font> or  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c'</font> )
 ;<br/>

<a NAME="E2:29_1_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b1</font>,<font color="Maroon">a</font> or  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:29_1_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/>

<a NAME="E2:29_1_1_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 ;<br/>

<a NAME="E2:29_1_1_3_1_1"/><i><font color="Green">E82</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:29_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:29_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E4:29_1_1_3_1_1_1"/>
<font color="Maroon">o</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_3_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1_3_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_3_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 ;<br/>

<a NAME="E2:29_1_1_3_1_2"/><i><font color="Green">E85</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_1_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:29_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:29_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E4:29_1_1_3_1_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_3_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_3_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1_3_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_1_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1_1_3"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/>

<a NAME="E2:29_1_1_3_2"/><i><font color="Green">E88</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_3_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:29_1_1_3_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:29_1_1_3_2_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E4:29_1_1_3_2_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E5:29_1_1_3_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_3_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29_1_1_3_2"/><i><font color="Green">E90</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a1</font>
 
<b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E4:29_1_1_3_2"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a1</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1_1_3_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:29_1_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E6:29_1_1_3"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">M</font> or <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E7:29_1_1_3"/>
( <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T66">AFF_1:66</a></i>;<br/>
<a NAME="E8:29_1_1_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> or <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/>
<b>hence </b><a NAME="E9:29_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:29_1_1"/>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E9:29_1_1"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i>, , , , , <a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/><b>hence </b><a NAME="E10:29_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1"/>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29"/>
(  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  implies  <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E93</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:29_2_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">o'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1'</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">b'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">a'</font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">c'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a'</font>,<font color="Maroon">a1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b'</font>,<font color="Maroon">b1'</font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c'</font>,<font color="Maroon">c1'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1'</font>,<font color="Maroon">c1'</font> )
 ;<br/><b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o'</font>, <font color="Maroon">a</font> = <font color="Maroon">a'</font>, <font color="Maroon">a1</font> = <font color="Maroon">a1'</font>, <font color="Maroon">b</font> = <font color="Maroon">b'</font>, <font color="Maroon">b1</font> = <font color="Maroon">b1'</font>, <font color="Maroon">c</font> = <font color="Maroon">c'</font>, <font color="Maroon">c1</font> = <font color="Maroon">c1'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E3:29_2_1"/><i><font color="Green">E95</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">b1</font>,<font color="Maroon">a</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>,<font color="Maroon">c</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a1</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b1</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">c1</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:29_2_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">A</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:29_2_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">M</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E9:29_2_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">N</font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E10:29_2_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">A</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">A</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E11:29_2_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1_1"/>
(  not <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> or  not <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E2:29_2_1_1"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">b1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> or <font color="Maroon">a</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:29_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:29_2_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E13:29_2_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 <b>by </b><i>, , , , , , <a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/><b>hence </b><a NAME="E14:29_2_1"/>
<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1'</font>,<font color="Maroon">c1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_2"/>
 <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="conaffm.html#D1">CONAFFM:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
(  <a href="conaffm.html#NR1" target="_self">DES_holds_in</a> <font color="Maroon">X</font> iff  <a href="analmetr.html#K3">Af</a> <font color="Maroon">X</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  )
 <b>by </b><i/>;<br/>


</div>
