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<a NAME="E1:3_1_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <font color="Olive">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:3_1_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">z</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Olive">v</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> ;<br/>
<a NAME="E4:3_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">z'</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/>
<a NAME="E5:3_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><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 <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><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> ;<br/><b>assume </b><a NAME="E2:3_1_1_1_1"/>
<font color="Maroon">b'</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:3_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 ;<br/><a NAME="E4:3_1_1_1_1"/>
<font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/><b>hence </b><a NAME="E5:3_1_1_1_1"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:3_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<b>take </b><font color="Maroon">x</font> =  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:3_1_1_1_2"/>
<font color="Maroon">a</font> <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="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 ;<br/><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> ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E4:3_1_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a'</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">x'</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/><a NAME="E5:3_1_1_1_2"/>
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D11">YELLOW_0:def 11</a></i>;<br/><a NAME="E6:3_1_1_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:3_1_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_1"/>
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font> is <a href="yellow_0.html#V1">lower-bounded</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a></i>;<br/>


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