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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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="closure2.html#NM3">SetOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V8" target="_self">monotonic</a>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="closure2.html#V8" target="_self">monotonic</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13" target="_self">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E3:65"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E4:65"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:65"/>
<font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="closure2.html#E2:65"><i><font color="Green">E30</font></i></a>, <a class="txt" href="closure2.html#E3:65"><i><font color="Green">E31</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" 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="closure2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:65"><i><font color="Green">E29</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" 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="closure2.html#K8" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="closure2.html#E4:65"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K8" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</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="closure2.html#K8" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:65"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


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