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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel35.html#V1" target="_self">extra-order</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  non <a href="xboole_0.html#V1">empty</a>  <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C</font> is <a href="waybel35.html#V4" target="_self">sup-closed</a>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Olive">c</font>,<font color="Maroon">L</font>
 ;<br/>

<b>assume </b><a NAME="E3:53"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R</font> <a href="waybel35.html#R1" target="_self">satisfies_SIC_on</a> <font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E4:53"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">d</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>;<br/>
<a NAME="E5:53"/><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:53"/>
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font> <a href="lattice7.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, <a class="txt" href="waybel35.html#E1:53"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E8:53"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span>
 
;<br/>
<a NAME="E9:53"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="txt" href="waybel35.html#E2:53"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E3:53_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c</font> <a href="orders_2.html#R2">&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 ;<br/><a NAME="E2:53_1_2_1"/><i><font color="Green">E68</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E3:53_1_2_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:53_1_2_1_1"/>
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:53_1_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E3:53_1_2_1_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_2_1_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel35.html#T2" target="_self">Th2</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:53_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel35.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:53_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:53_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E8:53_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E9:53_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="waybel35.html#E2:53"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel35.html#E3:53"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E10:53_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="waybel35.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E11:53_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/><b>hence </b><a NAME="E12:53_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel35.html#T2" target="_self">Th2</a>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
