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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L'</font> be    <a href="nat_lat.html#M2">SubLattice</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L'</font> is <a href="msualg_7.html#V2" target="_self">\/-inheriting</a>
 ;<br/>

<a NAME="E2:20"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L'</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> st <br/>( <font color="Olive">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">a</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">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<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>set </b><font color="Maroon">a</font> =  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> =  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">a'</font>
;<br/>

<a NAME="E2:20_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a'</font>
 <b>by </b><i/>;<br/>

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

<a NAME="E4:20_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</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="nat_lat.html#D16">NAT_LAT:def 16</a></i>;<br/>
<b>then 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> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><a NAME="E6:20_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">b'</font>
 ;<br/>

<a NAME="E7:20_1"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:20_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/>
<b>hence </b><a NAME="E10:20_1"/>
<font color="Maroon">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon">L'</font> is <a href="lattice3.html#V4">complete</a>
 <b>by </b><i/>;<br/>


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