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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V2">upper-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V2">auxiliary(ii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C</font> is <a href="waybel35.html#V2" target="_self">maximal</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</font> <a href="waybel_1.html#R4">is_maximum_of</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E4:37"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">C</font> <a href="lattice7.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:37_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E3:37_1"/><i><font color="Green">E64</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D7">WAYBEL_1:def 7</a></i>;<br/><a NAME="E4:37_1"/><i><font color="Green">E65</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">C</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D7">WAYBEL_1:def 7</a></i>;<br/><a NAME="E5:37_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is    <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel35.html#D3" target="_self">WAYBEL35:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E2:37_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:37_1_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E4:37_1_1_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/><b>hence </b><a NAME="E5:37_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , <a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, , <a class="ref" href="waybel_4.html#D5">WAYBEL_4:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:37_1_1_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:37_1_1_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E4:37_1_1_1_3"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/><b>hence </b><a NAME="E5:37_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , <a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, , <a class="ref" href="waybel_4.html#D5">WAYBEL_4:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_4"><b>suppose </b></a><a NAME="E1:37_1_1_1_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:37_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:37_1"/><i><font color="Green">E75</font></i>: 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:37_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="lattice7.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E9:37_1"/><b>then </b>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/><b>hence </b><a NAME="E10:37_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="waybel35.html#E1"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/>


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