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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:11_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="msualg_5.html#E2:11_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="msualg_5.html#E1:11_1_1_1"><i><font color="Green">E8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E4:11_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_4.html#D2">MSUALG_4:def 2</a></i>;<br/>
<b>take </b>
 <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E5:11_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_5.html#E2:11_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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