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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> )
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c5">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 ;<br/>

<font color="Maroon" title="c5">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c6">B</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="msualg_7.html#E1:8_1"><i><font color="Green" title="E6">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="E3:8_1"/>
<font color="Maroon" title="c5">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#T17" title="PBOOLE:th.17">PBOOLE:17</a></i>;<br/>


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

<b>thus </b><a NAME="E3:8"/>
( <font color="Maroon" title="c5">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="msualg_7.html#E1:8_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="pboole.html#T25" title="PBOOLE:th.25">PBOOLE:25</a></i>
;<br/>
<b>hence </b><a NAME="E3:8_2"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</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>


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