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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> 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="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>;<br/>









<a NAME="E1:29"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T12">MSSUBFAM:12</a></i>;<br/>
<a NAME="E2:29"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E3:29"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T16">MBOOLEAN:16</a></i>;<br/>
<a NAME="E4:29"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T11">MSSUBFAM:11</a></i>;<br/>
<b>assume </b><a NAME="E6:29"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="closure1.html#V4" target="_self">topological</a>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="closure1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T73">PBOOLE:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E5:29"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure1.html#E6:29"><i><font color="Green">E13</font></i></a>, <a class="ref" href="closure1.html#D5" target="_self">Def5</a></i>
;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T59">PBOOLE:59</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T81">PBOOLE:81</a></i>;<br/>
<a NAME="E11:29"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T62">PBOOLE:62</a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K4">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E10:29"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>


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