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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V3">idempotent</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V2">associative</a> )
 ;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ;<br/>

<a NAME="E3:47"/><i><font color="Green">E33</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="setwiseo.html#T26" target="_self">Th26</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setwiseo.html#T23" target="_self">Th23</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="setwiseo.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="setwiseo.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="setwiseo.html#E1:47_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="setwiseo.html#T40" target="_self">Th40</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="setwiseo.html#K2" target="_self">.}</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="setwiseo.html#T29" target="_self">Th29</a></i>;<br/>


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