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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</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">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:41_1_1"/><i><font color="Green">E20</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; ( for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon">f</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:41_1_1_1"/>
<font color="Maroon">x</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">x</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:41_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">f</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/>;<br/>
<b>reconsider </b><font color="Maroon">u0</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E5:41_1_1_1"/>
( <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">u0</font> &amp;  <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">u0</font> &amp; <font color="Maroon">u0</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:41_1_1_1"/><b>then </b>
<font color="Maroon">u0</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/>;<br/>
<a NAME="E7:41_1_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</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="E8:41_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:41_1_1"/>
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</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="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">A</font>,<font color="Olive">s</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
