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<b>let </b><font color="Maroon">B1</font> be    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>, <font color="Maroon">B2</font> be    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">B1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">ff</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<a NAME="E3:13_1_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">B1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then </b><font color="Maroon">B1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:13_1_2_1"/>
<font color="Maroon">B1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_1_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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