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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V6">join-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E2:7"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">V</font>,<font color="Maroon">S</font></span>)</span> where <font color="Olive">V</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">S</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:7"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E7:7"/><i><font color="Green">E36</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z</font>,<font color="Maroon">S</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
(  not <font color="Maroon">Z</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Z</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1"/>
not <font color="Maroon">Z</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:7_1_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>reconsider </b><font color="Maroon">xl</font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel21.html#T15">WAYBEL21:15</a></i>;<br/><a NAME="E5:7_1_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z1</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z1</font> holds <br/> <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/><a NAME="E6:7_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">xl</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">b1</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><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>assume </b><a NAME="E2:7_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/>

<a NAME="E3:7_1_1_1_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">xl</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>reconsider </b><font color="Maroon">b1</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><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:7_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z1</font>
 ;<br/><a NAME="E3:7_1_1_1_2"/>
<font color="Maroon">b1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c1</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="E1:7_1_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:7_1_1_1_2_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_2_1"/>
<font color="Maroon">c1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:7_1_1_1_2"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_1_2"/>
<font color="Maroon">xl</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div><a NAME="E8:7_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/><a NAME="E9:7_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">V</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">V</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">L</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Z</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:7_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">dS</font> =  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>reconsider </b><font color="Maroon">dL</font> =  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E5:7_1_1_2"/>
<font color="Maroon">S</font> is <a href="yellow_0.html#V1">lower-bounded</a>
 <b>by </b><i>, , <a class="ref" href="waybel20.html#T6">WAYBEL20:6</a></i>;<br/><a NAME="E6:7_1_1_2"/><b>then </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">dL</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E7:7_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">dS</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><a NAME="E8:7_1_1_2"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">dS</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E9:7_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">dS</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/><a NAME="E10:7_1_1_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E11:7_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">V</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">V</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">L</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E12:7_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">X</font>
 ;<br/>


</div>
