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

<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">OU0</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font>;<br/>





<a NAME="E1:39"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">B</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></i>;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> iff <font color="Maroon" title="c3">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#T15" title="MSUALG_2:th.15">MSUALG_2:15</a></i>;<br/>
<b>thus </b><a NAME="E3:39"/>
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> implies <font color="Maroon" title="c3">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">B1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">B1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> )
 ;<br/>
<b>thus </b><a NAME="E4:39_1"/>
<font color="Maroon" title="c3">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:39_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="msualg_2.html#T15" title="MSUALG_2:th.15">MSUALG_2:15</a></i>;<br/>


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

<b>assume </b><a NAME="E4:39"/>
<font color="Maroon" title="c3">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 ;<br/>

<b>hence </b><a NAME="E5:39"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E1:39"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="osalg_2.html#E2:39"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>


</div>
