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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">MA</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<a NAME="E1:27"/>
 <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">MA</font> <a href="tarski.html#R1">c=</a>  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_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="E1:27_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">MA</font>
 ;<br/>

<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D12">MSUALG_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1">CLOSURE2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SUBMA</font> =  <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">MA</font> as   <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font> ;<br/>
<a NAME="E3:27"/>
for <font color="Olive">F</font> being  <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>  st <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SUBMA</font> holds <br/> <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Olive">F</font></span><a href="closure2.html#K5">:|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">SUBMA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>;<br/>

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SUBMA</font>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span>;<br/>
<b>set </b><font color="Maroon">M</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">MA</font></span>)</span>;<br/>
<a NAME="E2:27_2"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <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="E3:27_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></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">MA</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:27_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:27_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E6:27_2_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:27_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/>
<a NAME="E8:27_2_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</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="E9:27_2_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">MA</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:27_2_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">MA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:27_2_1"/><b>then </b>
<font color="Maroon">u</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">MA</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:27_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</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">MA</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E13:27_2_1"/>
<font color="Maroon">u</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">MA</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> <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">MA</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">MA</font> ;<br/>
<a NAME="E6:27_2"/>
for <font color="Olive">B</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">MA</font>  st <font color="Olive">B</font> <a href="pboole.html#R6">=</a> <font color="Maroon">x</font> holds <br/><font color="Olive">B</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:27_2"/>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">SUBMA</font>
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D12">MSUALG_2:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
 <a href="msualg_2.html#K6">SubSort</a> <font color="Maroon">MA</font> is  <a href="closure2.html#V4">absolutely-multiplicative</a> <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MA</font>
 <b>by </b><i><a class="ref" href="closure2.html#D8">CLOSURE2:def 8</a></i>;<br/>


</div>
