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<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/>





<b>thus </b><a NAME="E1:52"/>
 <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <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="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:52_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1"/>
 <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>


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

<a NAME="E2:52"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E3:52"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E2:52"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>


</div>
