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<b>let </b><font color="Maroon" title="c1">L</font> be   <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>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>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c5">a'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>;<br/>









<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font>
 ;<br/>

<b>thus </b><a NAME="E2:18"/>
( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">a</font> implies <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">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" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">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" title="c6">q'</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="E1:18_1_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">q</font> = <font color="Maroon" title="c6">q'</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="E3:18_1_1"/>
<font color="Maroon" title="c6">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><a NAME="E4:18_1_1"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:18_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><font color="Maroon" title="c6">q'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a'</font> = 
<font color="Maroon" title="c7">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c6">q'</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E4:18_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E6:18_1_1"/>
<font color="Maroon" title="c6">q'</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="E3:18_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>


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

<b>thus </b><a NAME="E3:18"/>
( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">a'</font> implies <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">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" title="E15">A4</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">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" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">q'</font> = <font color="Maroon" title="c6">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> ;<br/><a NAME="E3:18_2_1"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c7">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:18_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="msualg_7.html#E1:18_2_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><font color="Maroon" title="c6">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c7">q'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a'</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E3:18_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E5:18_2_1"/>
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">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="E3:18_2"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>


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


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