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<b>set </b><font color="Maroon">C</font> =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">B</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:40_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:40_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E4:40_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">B</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_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> ;<br/>

<b>thus </b><a NAME="E1:40_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">B</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K5" target="_self">SubSort</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">B</font> being   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font><b> such that </b><a NAME="E2:40_1_1_1"/><i><font color="Green">E26</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E3:40_1_1_1"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 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="E4:40_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="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
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="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E6:40_1_1_1"/>
<font color="Maroon">B</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</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="E7:40_1_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E8:40_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:40_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="msualg_2.html#E2:40_1_1_1"><i><font color="Green">E26</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<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="E6:40_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="E7:40_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><a class="txt" href="msualg_2.html#E4:40_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E8:40_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> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E10:40_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</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="E11:40_1_1"/>
not  <a href="msualg_2.html#K8" target="_self">SubSort</a> <font color="Maroon">A</font>,<font color="Maroon">s</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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