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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/>









<b>assume </b><a NAME="E1:27"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="closure1.html#V2" target="_self">monotonic</a>
 ;<br/>

<a NAME="E2:27"/>
<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="E3:27"/><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="E4:27"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</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#T15">MBOOLEAN:15</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</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="E6:27"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</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/>
<a NAME="E7:27"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T17">PBOOLE:17</a></i>;<br/>
<a NAME="E8:27"/><b>then </b><i><font color="Green">E14</font></i>: 
<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#K3">/\</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#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:27"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure1.html#E6:27"><i><font color="Green">E13</font></i></a>, <a class="ref" href="closure1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E9:27"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T17">PBOOLE:17</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
<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#K3">/\</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#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:27"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure1.html#E6:27"><i><font color="Green">E13</font></i></a>, <a class="ref" href="closure1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E11:27"/>
<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#K3">/\</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#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</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#E8:27"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#T19">PBOOLE:19</a></i>;<br/>


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