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









<b>assume </b><a NAME="E1:18"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<b>thus </b><a NAME="E2:18"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font> implies <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><a NAME="E1:18_1_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">q</font> = <font color="Maroon">q'</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="E3:18_1_1"/>
<font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E4:18_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><font color="Maroon">q'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a'</font> = 
<font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">q'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E6:18_1_1"/>
<font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</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="E3:18_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:18"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a'</font> implies <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/><a NAME="E3:18_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
<font color="Maroon">q'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E5:18_2_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</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="E3:18_2"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>


</div><b>end;</b></div>


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