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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:39"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:39"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>3</sub></font> is <a href="msualg_2.html#V3">opers_closed</a> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#T15">MSUALG_2:15</a></i>;<br/>
<b>thus </b><a NAME="E3:39"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K6" target="_self">OSSubSort</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is <a href="msualg_2.html#V3">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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K6" target="_self">OSSubSort</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:39_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_2.html#T15">MSUALG_2:15</a></i>;<br/>


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

<b>assume </b><a NAME="E4:39"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 ;<br/>

<b>hence </b><a NAME="E5:39"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K6" target="_self">OSSubSort</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E1:39"><i><font color="Green">E22</font></i></a>, <a class="txt" href="osalg_2.html#E2:39"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
