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<div class="add">

<b>let </b><font color="Maroon">DP</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_3.html#V1">discrete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><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">OAF</font> be    <a href="msalimit.html#M1" target="_self">OrderedAlgFam</a> of <font color="Maroon">DP</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="msalimit.html#V1" target="_self">normalized</a>  <a href="msalimit.html#M2" target="_self">Binding</a> of <font color="Maroon">OAF</font>;<br/>







<a NAME="E1:15"/>
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">a</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">a</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<b>thus </b><a NAME="E3:15_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:15_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:15_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E3:15_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:15_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <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/>
<b>hence </b><a NAME="E5:15_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">e</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="E4:15_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i><a class="ref" href="pralg_2.html#T20">PRALG_2:20</a></i>;<br/>
<a NAME="E6:15_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DP</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DP</font>, <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DP</font>;<br/>

<b>assume </b><a NAME="E1:15_1_2"/>
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:15_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#T1">ORDERS_3:1</a></i>;<br/>
<a NAME="E3:15_1_2"/>
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<b>then </b> <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">i</font> = 
<font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
 <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E5:15_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/>
<a NAME="E6:15_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">DP</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:15_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
;<br/>
<a NAME="E8:15_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E9:15_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E11:15_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<b>thus </b><a NAME="E12:15_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:15_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15"/><b>then </b><i><font color="Green">E47</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<a NAME="E3:15"/>
 <a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<b>hence </b><a NAME="E4:15"/>
 <a href="msalimit.html#K3" target="_self">InvLim</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font>
 <b>by </b><i>, <a class="ref" href="msualg_2.html#T10">MSUALG_2:10</a></i>;<br/>


</div>
