<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font>, <font color="Maroon">B</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font>;<br/>





<b>reconsider </b><font color="Maroon">MAB</font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="closure3.html#K4" target="_self">/\</a> <font color="Maroon">B</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">MA</font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">MB</font> =  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E4:21"/>
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">MAB</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">MA</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">MB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">MAB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure3.html#M1" target="_self">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> <a href="closure3.html#K4" target="_self">/\</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">MA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure3.html#M1" target="_self">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>, </i>;<br/>
<a NAME="E4:21_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">MB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure3.html#M1" target="_self">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">B</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:21_1"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MAB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MA</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">MB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MAB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:21_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</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="closure3.html#M1" target="_self">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> <a href="closure3.html#K4" target="_self">/\</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E6:21_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:21_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="closure3.html#K4" target="_self">/\</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:21_1_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:21_1_1"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</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="closure3.html#M1" target="_self">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>  &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</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="closure3.html#M1" target="_self">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">B</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<a NAME="E10:21_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</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="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure3.html#M1" target="_self">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>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</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="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure3.html#M1" target="_self">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">B</font> </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:21_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">MB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:21_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MA</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">MB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_1"/>
<font color="Maroon">MAB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">MA</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">MB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="closure3.html#K4" target="_self">/\</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


</div>
