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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">L'</font> be    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a>
 ;<br/>

<a NAME="E2:19"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L'</font>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>  st <font color="Olive" title="b3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L'</font>;<br/>

<b>set </b><font color="Maroon" title="c4">a</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a'</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> <b>by </b><i><a class="txt" href="msualg_7.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="msualg_7.html#D2" target="_self" title="MSUALG_7:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a'</font>
;<br/>

<a NAME="E2:19_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon" title="c5">a'</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T13" target="_self" title="MSUALG_7:th.13">Th13</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>;<br/>

<a NAME="E4:19_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="nat_lat.html#D16" title="NAT_LAT:def.16">NAT_LAT:def 16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><a NAME="E6:19_1"/>
<font color="Maroon" title="c6">b'</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E7:19_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="msualg_7.html#T13" target="_self" title="MSUALG_7:th.13">Th13</a></i>;<br/>
<a NAME="E8:19_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40" title="LATTICE3:th.40">LATTICE3:40</a></i>;<br/>
<font color="Maroon" title="c6">b'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a'</font> = 
<font color="Maroon" title="c7">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c6">b'</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E8:19_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
;<br/>
<b>hence </b><a NAME="E10:19_1"/>
<font color="Maroon" title="c6">b'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon" title="c2">L'</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D6" title="VECTSP_8:def.6">VECTSP_8:def 6</a></i>;<br/>


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