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<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">OU0</font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>, <font color="Maroon">B</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>;<br/>





<a NAME="E1:38"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">B</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:38"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K5" target="_self">OSSubSort</a> <font color="Maroon">A</font> implies ( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp;  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K5" target="_self">OSSubSort</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K5">SubSort</a> <font color="Maroon">A</font> : <font color="Olive">x</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">B1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K5">SubSort</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B1</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> )
 ;<br/>
<a NAME="E5:38_1"/>
( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp;  <a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#T14">MSUALG_2:14</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1"/>
( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp;  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:38"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp;  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E4:38"/>
 <a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38"/><b>then </b>
 <a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E6:38"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5">SubSort</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#T14">MSUALG_2:14</a></i>;<br/>
<a NAME="E7:38"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K5">SubSort</a> <font color="Maroon">A</font> : <font color="Olive">x</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:38"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K5" target="_self">OSSubSort</a> <font color="Maroon">A</font>
 ;<br/>


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