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<a NAME="E1:51_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">U0</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="msualg_2.html#K9" target="_self">MSSubSort</a> <font color="Maroon">A</font> as  <a href="relat_1.html#V2">V5</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/>
<a NAME="E3:51_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">a</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:51_1_1"/><b>then </b>
<font color="Maroon">U0</font> <a href="msualg_2.html#K10" target="_self">|</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">a</font></span>)</span> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">U0</font> <a href="msualg_2.html#K10" target="_self">|</a> <font color="Maroon">a</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:51_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4" target="_self">Opers</a> <font color="Maroon">U0</font>,<font color="Maroon">a</font></span>)</span> #)
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:51_1_1_1"/>
<font color="Maroon">A</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">u1</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><b>let </b><font color="Maroon">U2</font> be    <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">U0</font>;<br/><b>reconsider </b><font color="Maroon">B</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i/>;<br/><a NAME="E4:51_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">B</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E5:51_1_1_1"/>
<font color="Maroon">A</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U2</font>
 ;<br/><a NAME="E6:51_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<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#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:51_1_1_1"/>
 <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">U0</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U2</font>
 <b>by </b><i/>;<br/><a NAME="E8:51_1_1_1"/><b>then </b>
 <a href="msualg_2.html#K2" target="_self">Constants</a> <font color="Maroon">U0</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E9:51_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="txt" href="msualg_2.html#E4:51_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E10:51_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">u1</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1"><b>proof </b></a><div class="add">

<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="E1:51_1_1_1_1"/>
<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">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:51_1_1_1_1"/><i><font color="Green">E28</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">u1</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="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:51_1_1_1_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">u1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:51_1_1_1"/>
<font color="Maroon">u1</font> is    <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">U2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:51_1_1"/>
 <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">A</font> is <a href="msualg_1.html#V5">non-empty</a>
 <b>by </b><i/>;<br/>


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