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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a> ) );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:35_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:35_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:35_1_1"/>
 <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E5:35_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T100">ZFMISC_1:100</a></i>;<br/>
<a NAME="E6:35_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E3:35_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:35_1_1"/><i><font color="Green">E25</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E8:35_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="msualg_2.html#E2:35_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="msualg_2.html#E6:35_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msualg_2.html#E7:35_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E10:35_1_1"/>
 <a href="msualg_2.html#K6" target="_self">SubSort</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E2:35_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_2.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E11:35_1_1"/>
not  <a href="msualg_2.html#K6" target="_self">SubSort</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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