<?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">AA</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> holds <br/><font color="Olive">x</font> is   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</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>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Olive">X</font></span>)</span> where <font color="Olive">X</font> is   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font> : <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">AA</font>
 ;<br/>

<b>thus </b><a NAME="E4:22"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_1"/>
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/><span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <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"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</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="22_1_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="E1:22_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:22_1_1_1_1"/><b>then </b>
<font color="Maroon">a</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>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:22_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</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><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</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="E7:22_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:22_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E10:22_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="pboole.html#R6">=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E11:22_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">Q</font></span>)</span> <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">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">g</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">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:22_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E15:22_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">g</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">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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="E17:22_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">d</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="E18:22_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<a NAME="E19:22_1_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">AA</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E20:22_1_1_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">h</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">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">AA</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:22_1_1_1_1"/><b>then </b>
<font color="Maroon">a</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">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">h</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">h</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">AA</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E22:22_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <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"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</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="E2:22_1"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E5:22"/>
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/><span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <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"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</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="22_2"><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="E1:22_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<a NAME="E2:22_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</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="22_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:22_2_1"/><b>then </b>
<font color="Maroon">a</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> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:22_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y</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> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</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="E7:22_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:22_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:22_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E11:22_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<a NAME="E13:22_2_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c</font></span>)</span> <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">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">v</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">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:22_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">v</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">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E15:22_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</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">v</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">v</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">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E16:22_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Olive">X</font></span>)</span> where <font color="Olive">X</font> is   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font> : <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">cos</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:22_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cos</font>
 <b>and </b><br/><a NAME="E19:22_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">cos</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E20:22_2_1"/>
<font color="Maroon">cos</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">t</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">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E21:22_2_1"/><b>then </b>
<font color="Maroon">a</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">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">t</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">t</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/>
<b>hence </b><a NAME="E22:22_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_2"/>
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


</div>
