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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">U0</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>;<br/>





<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K6" target="_self">SubSort</a> <font color="Maroon">U0</font> implies <font color="Maroon">B</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a> )
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:38"><i><font color="Green">E21</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">B</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 ;<br/>

<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/>
<a NAME="E3:38"/><i><font color="Green">E21</font></i>: 
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:38"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="msualg_2.html#E5:38_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:38_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="E7:38_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">y</font>
 
<b>by </b><i>, <a class="txt" href="msualg_2.html#E7:38_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E8:38_1"/>
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">y</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="txt" href="msualg_2.html#E7:38_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="txt" href="msualg_2.html#E5:38_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="msualg_2.html#E7:38_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msualg_2.html#E8:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:38"/><b>then </b>
 <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> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/>
<a NAME="E6:38"/><b>then </b>
(  <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> <font color="Maroon">B</font></span>)</span></span>)</span> <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">U0</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</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> <font color="Maroon">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="zfmisc_1.html#T100">ZFMISC_1:100</a></i>;<br/>
<a NAME="E7:38"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">B</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">U0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:38"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">B</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">S</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">U0</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38"/>
for <font color="Olive">C</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>  st <font color="Olive">C</font> <a href="pboole.html#R6">=</a> <font color="Maroon">B</font> holds <br/><font color="Olive">C</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:38"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K6" target="_self">SubSort</a> <font color="Maroon">U0</font>
 <b>by </b><i>, <a class="txt" href="msualg_2.html#E3:38"><i><font color="Green">E21</font></i></a></i>;<br/>


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