<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span> <b>by </b><i><a class="ref" href="msualg_7.html#T6" target="_self">Th6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> =  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E2:12_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>thus </b><a NAME="E5:12_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T45">LATTICES:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E9</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_7.html#T9" target="_self">Th9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><b>assume </b><a NAME="E2:12_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:12_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:12_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:12_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><b>assume </b><a NAME="E2:12_1_1_2_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_2_2_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span> <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E3:12_1_1_2_2_1"/>
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="closure2.html#K5">:|</a></span> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/><a NAME="E4:12_1_1_2_2_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:12_1_1_2_2_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_2_2_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:12_1_1_2_2_1_1"/>
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> where B is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="closure2.html#T15">CLOSURE2:15</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span><b> such that </b><br/><a NAME="E4:12_1_1_2_2_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2_2_1_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="msualg_7.html#E4:12_1_1_2_2_1_1"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><a NAME="E7:12_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E2:12_1_1_2_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msualg_7.html#E4:12_1_1_2_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><a NAME="E8:12_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E9:12_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:12_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_7.html#E4:12_1_1_2_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E8:12_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_7.html#E4:12_1_1_2_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/><b>hence </b><a NAME="E9:12_1_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E4:12_1_1_2_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msualg_7.html#E6:12_1_1_2_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E4:12_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="closure2.html#K5">:|</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E5:12_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:12_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12"/>
 <a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V4">complete</a>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D6">VECTSP_8:def 6</a></i>;<br/>


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