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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>assume </b><a NAME="E1:141_3_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">L</font> is <a href="waybel_0.html#V24" target="_self">up-complete</a> &amp; <font color="Maroon">L</font> is <a href="lattice3.html#V1">with_suprema</a> &amp; <font color="Maroon">L</font> is <a href="yellow_0.html#V1">lower-bounded</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V1">with_suprema</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">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a><br/>

<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">X'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">K</font> ;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/><a NAME="E1:141_3_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E2:141_3_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_3_1_2"><b>now </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>;<br/><b>assume </b><a NAME="E1:141_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">X'</font>
 ;<br/><a NAME="E2:141_3_1_2"/><b>then </b>
 ex <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X'</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> )
 ;<br/><b>hence </b><a NAME="E3:141_3_1_2"/>
 ex <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X'</font> st <br/>(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_3_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>assume </b><a NAME="E2:141_3_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:141_3_1_3"/><b>then </b>
 <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="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/><b>hence </b><a NAME="E4:141_3_1_3"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">X'</font>
 ;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">fX</font> =  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1" target="_self">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:141_3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">fX</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E11:141_3_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">fX</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<a NAME="E12:141_3_1"/>
<font color="Maroon">X'</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_0.html#D13" target="_self">Def13</a>, </i>;<br/>
<b>hence </b><a NAME="E13:141_3_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T5">YELLOW_0:5</a></i>;<br/>

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

<b>assume </b><a NAME="E14:141_3_1"/>
<font color="Maroon">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E15:141_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T5">YELLOW_0:5</a></i>;<br/>
<a NAME="E16:141_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">fX</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E17:141_3_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D14" target="_self">Def14</a></i>;<br/>


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