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<b>let </b><font color="Maroon">S</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U0</font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U1</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">U1</font> is   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U0</font>
 ;<br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green">E26</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/><a NAME="E3:15_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:15_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font>;<br/><b>assume </b><a NAME="E5:15_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 ;<br/><b>thus </b><a NAME="E6:15_1"/>
( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E45</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font>
 ;<br/>

<a NAME="E3:15"/><b>then </b><i><font color="Green">E46</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:15"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">B</font> being   <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font>  st <font color="Olive">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> holds <br/>( <font color="Olive">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U0</font>,<font color="Olive">B</font> )
 ;<br/>

<a NAME="E5:15"/>
<font color="Maroon">U1</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">U0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a><br/>

<b>let </b><font color="Maroon">B</font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>;<br/>

<b>assume </b><a NAME="E2:15_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:15_2"/>
( <font color="Maroon">B1</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:15_2"/>
( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">U1</font> is   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U0</font>
 <b>by </b><i>, <a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>


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