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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V1">auxiliary(i)</a> <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="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds   <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>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel35.html#D9" target="_self">WAYBEL35:def 9</a><br/>

<b>set </b><font color="Maroon">s</font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<a NAME="E2:58"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>assume </b><a NAME="E3:58"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E4:58"/><i><font color="Green">E46</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>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:58"/>
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green">E60</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>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2:58"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E7:58"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</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>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</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">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E3:58_1"/><i><font color="Green">E65</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">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:58_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:58"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<a NAME="E5:58_1"/><b>then </b>
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">x</font> <a href="lattice7.html#R1">c=</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:58_1"/>
<font color="Maroon">x</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>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:58"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="waybel35.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:58"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</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>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:58"><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="E9:58"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</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>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E3:58"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<a NAME="E10:58"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E11:58"/><b>then </b>
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:58"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="waybel35.html#K6" target="_self">SupBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="waybel35.html#E1:58"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>;<br/>


</div>
