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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>2</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E31</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V3">idempotent</a> )
 <b>and </b><br/><a NAME="E2:49"/><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/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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>1</sub></font>;<br/>



<b>assume </b><a NAME="E3:49"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:49_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:49_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:49_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="setwiseo.html#E2:49"><i><font color="Green">E32</font></i></a>, <a class="ref" href="setwiseo.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E5:49_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
<font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="setwiseo.html#E3:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setwiseo.html#T35" target="_self">Th35</a></i>;<br/>


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