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

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

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E48</font></i>: 
 <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">K'</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="aff_2.html#D7">AFF_2:def 7</a><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>, <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>, <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>, <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>, <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>, <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>, <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="E2:9"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">K'</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">K'</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">c1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; not <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</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; <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> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K'</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="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">K'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green">E50</font></i>: 
( <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:9_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> implies ( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <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="9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:9_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> or <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">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green">E76</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="E2:9_1_1_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E3:9_1_1_1"/><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">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/><a NAME="E4:9_1_1_1"/><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">b1'</font>
 <b>by </b><i>, , <a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E5:9_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">c'</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="E6:9_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">b1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E7:9_1_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E8:9_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_2"/><i><font color="Green">E78</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="E2:9_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">o'</font>,<font color="Maroon">c1'</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#T13">AFF_1:13</a></i>;<br/><a NAME="E3:9_1_1_2"/>
<font color="Maroon">K'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E4:9_1_1_2"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c1'</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#T53">AFF_1:53</a></i>;<br/><a NAME="E5:9_1_1_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="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E6:9_1_1_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="E7:9_1_1_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="E8:9_1_1_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/><b>hence </b><a NAME="E9:9_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/><a NAME="E2:9_1_2"/><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="analmetr.html#T51">ANALMETR:51</a></i>;<br/><b>hence </b><a NAME="E3:9_1_2"/>
<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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 ;<br/><a NAME="E2:9_1_3"/><i><font color="Green">E82</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">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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">o</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="9_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_3_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 ;<br/>

<div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 ;<br/><a NAME="E2:9_1_3_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">o'</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#T13">AFF_1:13</a></i>;<br/><a NAME="E3:9_1_3_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>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:9_1_3_1_1"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">a1'</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="E5:9_1_3_1_1"/><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="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E6:9_1_3_1_1"/><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="E7:9_1_3_1_1"/><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/><b>hence </b><a NAME="E8:9_1_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:9_1_3_1_2"/>
<font color="Maroon">o'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E3:9_1_3_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">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/><a NAME="E4:9_1_3_1_2"/><b>then </b>
<font color="Maroon">a1'</font>,<font color="Maroon">o'</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="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E5:9_1_3_1_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="E6:9_1_3_1_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="E7:9_1_3_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E8:9_1_3_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">a'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E9:9_1_3_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/><a NAME="E2:9_1_3_2"/>
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="9_1_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_3_2_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="E2:9_1_3_2_1"/><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#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:9_1_3_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<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#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E4:9_1_3_2_1"/>
<font color="Maroon">o'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>
<a NAME="E5:9_1_3_2_1"/><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">c'</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/>
<a NAME="E6:9_1_3_2_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">o'</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="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E7:9_1_3_2_1"/><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#T13">AFF_1:13</a></i>;<br/>
<a NAME="E8:9_1_3_2_1"/><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#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_3_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:9_1_3_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/><a NAME="E4:9_1_3_2"/><i><font color="Green">E92</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="9_1_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_3_2_2"/>
 <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="E2:9_1_3_2_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/>
<b>hence </b><a NAME="E3:9_1_3_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:9_1_3_2"/>
<font color="Maroon">K'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E6:9_1_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">c1'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E7:9_1_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">c1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E8:9_1_3_2"/><b>then </b>
<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1'</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T23">AFF_1:23</a></i>;<br/><b>hence </b><a NAME="E9:9_1_3_2"/>
<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_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1'</font>
 ;<br/><a NAME="E2:9_1_3_3"/><i><font color="Green">E94</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>,<font color="Maroon">b</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="9_1_3_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_3_3_1"/><i><font color="Green">E95</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>,<font color="Maroon">b</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/>

<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_3_1_1"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:9_1_3_3_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">a1</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E3:9_1_3_3_1_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">o'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:9_1_3_3_1_1"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">a1'</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#D1">AFF_1:def 1</a></i>;<br/><a NAME="E5:9_1_3_3_1_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">a1</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E6:9_1_3_3_1_1"/><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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E7:9_1_3_3_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</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="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E8:9_1_3_3_1_1"/>
<font color="Maroon">a'</font>,<font color="Maroon">b'</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#T41">AFF_1:41</a></i>;<br/><a NAME="E9:9_1_3_3_1_1"/><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">c</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E10:9_1_3_3_1_1"/><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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E11:9_1_3_3_1_1"/><b>then </b>
 <a href="analmetr.html#R5">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="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E12:9_1_3_3_1_1"/><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="analmetr.html#T55">ANALMETR:55</a></i>;<br/><b>hence </b><a NAME="E13:9_1_3_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3_3_1_2"/>
 <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="E2:9_1_3_3_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<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="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E3:9_1_3_3_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a'</font>,<font color="Maroon">a1'</font>,<font color="Maroon">o'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:9_1_3_3_1_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">a1'</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#D1">AFF_1:def 1</a></i>;<br/><a NAME="E5:9_1_3_3_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">a1</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E6:9_1_3_3_1_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="analmetr.html#T82">ANALMETR:82</a></i>;<br/><a NAME="E7:9_1_3_3_1_2"/><b>then </b>
 <a href="analmetr.html#R5">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="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E8:9_1_3_3_1_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="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E9:9_1_3_3_1_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">a'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E10:9_1_3_3_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_3_3_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:9_1_3_3"/>
<font color="Maroon">K'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E4:9_1_3_3"/><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">c1'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E5:9_1_3_3"/><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">c1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E6:9_1_3_3"/><b>then </b><i><font color="Green">E100</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> )
 <b>by </b><i>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a>, <a class="ref" href="analmetr.html#D11">ANALMETR:def 11</a></i>;<br/><a NAME="E7:9_1_3_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/><a NAME="E8:9_1_3_3"/>
<font color="Maroon">K'</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><a NAME="E9:9_1_3_3"/><b>then </b>
<font color="Maroon">o'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E10:9_1_3_3"/><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">c'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/><a NAME="E11:9_1_3_3"/><b>then </b>
<font color="Maroon">b'</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#T13">AFF_1:13</a></i>;<br/><a NAME="E12:9_1_3_3"/><b>then </b>
<font color="Maroon">b</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="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E13:9_1_3_3"/><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>, , , , , , , </i>;<br/><b>hence </b><a NAME="E14:9_1_3_3"/>
<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="E5:9_1_3"/>
<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/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:9_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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:9"/>
<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_1.html#T49">AFF_1:49</a></i>;<br/>


</div>
