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<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/><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>let </b><font color="Maroon">c<sub>4</sub></font> be   <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>;<br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">c<sub>2</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>assume </b><a NAME="E3:14"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<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/>

<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">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="E2:14_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="E3:14_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="txt" href="msualg_7.html#E1:14"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msualg_7.html#E3:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_7.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:14_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>
<a NAME="E5:14"/><b>then </b><i><font color="Green">E11</font></i>: 
<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="14_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="E2:14_2"/><i><font color="Green">E12</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/><b>reconsider </b><font color="Maroon">c<sub>10</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#E1:14"><i><font color="Green">E9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="closure2.html#K5">:|</a></span> as  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</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> ;<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">c<sub>12</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:14_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#R3">in</a> <font color="Maroon">c<sub>11</sub></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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_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> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:14_2_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <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>13</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/><a NAME="E3:14_2_1_1"/>
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</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>13</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#E1:14_2_1_1"><i><font color="Green">E14</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="E5:14_2_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <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>13</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#E2:14_2_1_1"><i><font color="Green">E15</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#E1:14"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green">E16</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="E8:14_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#E1:14"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msualg_7.html#E2:14_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><a NAME="E9:14_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="E10:14_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E1:14_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msualg_7.html#E5:14_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E6:14_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="mssubfam.html#T45">MSSUBFAM:45</a></i>;<br/><b>hence </b><a NAME="E7:14_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="txt" href="msualg_7.html#E3:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_7.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E3:14"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msualg_7.html#E5:14"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>


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