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<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>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>5</sub></font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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="E1:71_1"/><i><font color="Green">E16</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="E3:71_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</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="E4:71_1"/><i><font color="Green">E18</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>7</sub></font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:71_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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>by </b><i><a class="txt" href="mssubfam.html#E1:71"><i><font color="Green">E14</font></i></a>, <a class="txt" href="mssubfam.html#E1:71_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/><a NAME="E6:71_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E3:71_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E7:71_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font></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#E2:71"><i><font color="Green">E15</font></i></a>, <a class="txt" href="mssubfam.html#E1:71_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/><a NAME="E8:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pboole.html#T5">PBOOLE:5</a></i>;<br/><a NAME="E9:71_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</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>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E10:71_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</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>by </b><i><a class="txt" href="mssubfam.html#E3:71_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssubfam.html#E5:71_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="setfam_1.html#T9">SETFAM_1:9</a></i>;<br/><a NAME="E11:71_1"/><b>then </b>
<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#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:71_1"/><b>then </b>
<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#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T5">PBOOLE:5</a></i>;<br/><b>hence </b><a NAME="E13:71_1"/>
<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="pboole.html#K3">/\</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> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font></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#E1:71_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="mssubfam.html#E4:71_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mssubfam.html#E6:71_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:71"/>
<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="pboole.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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