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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>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>







<b>assume </b><a NAME="E1:47"/>
<font color="Maroon">i</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">i</font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> ;<br/>
<a NAME="E3:47"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">A</font></span>)</span> <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/>
<a NAME="E4:47"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E26</font></i>: 
( <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> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:47_1"/>
(  <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>, </i>;<br/>
<a NAME="E5:47_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="pboole.html#K2">\/</a> <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="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E6:47_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="osalg_2.html#D1" target="_self">Def1</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


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