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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">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">X1</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span>;<br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; not <font color="Maroon" title="c3">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<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>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font> as   <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> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/>
<b>assume </b><a NAME="E3:14"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">q</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>reconsider </b><font color="Maroon" title="c9">q'</font> = <font color="Maroon" title="c8">q</font> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</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/><b>assume </b><a NAME="E2:14_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><a NAME="E3:14_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="msualg_7.html#E3:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="msualg_7.html#T8" target="_self" title="MSUALG_7:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c7">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" target="_self" title="MSUALG_7:th.7">Th7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:14"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c7">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="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">c</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>reconsider </b><font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c8">c</font> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</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/><b>assume </b><a NAME="E2:14_2"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">X1'</font> = <font color="Maroon" title="c4">X1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <b>by </b><i><a class="txt" href="msualg_7.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">S</font> = <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">X1'</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">Z1</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>assume </b><a NAME="E1:14_2_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c12">Z1</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> <font color="Maroon" title="c11">S</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_2_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:14_2_1_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c12">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14_2_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/><a NAME="E3:14_2_1_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">X1'</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span> where <font color="Olive" title="b1">x1</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> : <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14_2_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="closure2.html#T15" title="CLOSURE2:th.15">CLOSURE2:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">z</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span><b> such that </b><br/><a NAME="E5:14_2_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
( <font color="Maroon" title="c12">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X1</font> )
 <b>by </b><i><a class="txt" href="msualg_7.html#E2:14_2_1_1"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">z'</font> = <font color="Maroon" title="c14">z</font> as   <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> <b>by </b><i><a class="txt" href="msualg_7.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">z1</font> = <font color="Maroon" title="c15">z'</font> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</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/><a NAME="E8:14_2_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">z'</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="msualg_7.html#E2:14_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/><a NAME="E9:14_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">c'</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c16">z1</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" target="_self" title="MSUALG_7:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E10:14_2_1_1"/>
<font color="Maroon" title="c9">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14_2_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_2_1"/>
<font color="Maroon" title="c9">c'</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c12">Z1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E6:14_2"/><b>then </b>
<font color="Maroon" title="c9">c'</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>by </b><i><a class="ref" href="mssubfam.html#T45" title="MSSUBFAM:th.45">MSSUBFAM:45</a></i>;<br/><b>hence </b><a NAME="E7:14_2"/>
<font color="Maroon" title="c8">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a'</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E3:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="msualg_7.html#T7" target="_self" title="MSUALG_7:th.7">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon" title="c5">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E3:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="msualg_7.html#E5:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>


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