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<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>, <font color="Maroon">C</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="E1:6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="closure3.html#R1" target="_self">is_finer_than</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> <a href="closure3.html#R1" target="_self">is_finer_than</a> <font color="Maroon">C</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="closure3.html#D1" target="_self">CLOSURE3:def 1</a><br/>

<b>assume </b><a NAME="E3:6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </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="E8:6"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E10:6"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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