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

<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="closure3.html#R2" title="CLOSURE3:pred.2">is_coarser_than</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="closure3.html#R2" title="CLOSURE3:pred.2">is_coarser_than</a> <font color="Maroon" title="c5">C</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">CLOSURE3:def 2</a><br/>

<b>assume </b><a NAME="E3:6"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:6"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="closure3.html#E5:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="closure3.html#D2" target="_self" title="CLOSURE3:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">c</font>
;<br/>

<b>thus </b><a NAME="E10:6"/>
( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="closure3.html#E6:6"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="closure3.html#E8:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="closure3.html#E9:6"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div>
