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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:72"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:72_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:72_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:72_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="mssubfam.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:72_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E10:72_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="mssubfam.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E11:72_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E1:72"><i><font color="Green">E14</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="mssubfam.html#E3:72_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="mssubfam.html#E6:72_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mssubfam.html#E9:72_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_1"><b>case </b></a><a NAME="E1:72_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:72_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</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="mssubfam.html#E11:72_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="mssubfam.html#E10:72_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E11:72_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E4:72_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E7:72_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_2"><b>case </b></a><a NAME="E1:72_1_1_1_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E4:72_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssubfam.html#E10:72_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mssubfam.html#E11:72_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E7:72_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_3"><b>case </b></a><a NAME="E1:72_1_1_1_3"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E7:72_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssubfam.html#E10:72_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mssubfam.html#E11:72_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E4:72_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssubfam.html#E1:72_1_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_4"><b>case </b></a><a NAME="E1:72_1_1_1_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssubfam.html#E10:72_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mssubfam.html#E11:72_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mssubfam.html#E1:72_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="mssubfam.html#E4:72_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssubfam.html#E7:72_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:72_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:72"/>
 <a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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