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

<b>let </b><font color="Maroon">W1</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>, <font color="Maroon">W2</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>;<br/>

<b>assume </b><a NAME="E1:45_1_2"/><i><font color="Green">E47</font></i>: 
( ( for <font color="Olive">s</font> being  <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> holds  <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Olive">s</font></span>)</span> ) &amp; ( for <font color="Olive">s</font> being  <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> holds  <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Olive">s</font></span>)</span> ) )
 ;<br/>

<a NAME="E2:45_1_2"/>
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> holds <br/><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> ;<br/>
<a NAME="E3:45_1_2_1"/>
( <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:45_1_2_1"/>
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_2"/>
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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