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<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">OU0</font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">OU0</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> ;<br/><b>set </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">c2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> ;<br/><a NAME="E3:25_1"/>
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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span>  iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><b> such that </b><br/><a NAME="E3:25_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 ;<br/><a NAME="E4:25_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D5">MSUALG_2:def 5</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><b> such that </b><br/><a NAME="E4:25_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 ;<br/>
<a NAME="E5:25_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D5">MSUALG_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:25_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_2.html#K1">Constants</a> <font color="Maroon">OU0</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> : <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="osalg_2.html#K1" target="_self">OSConstants</a> <font color="Maroon">OU0</font>,<font color="Maroon">s</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E6:25_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:25"/><b>then </b>
( ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 
;<br/>
<a NAME="E3:25"/><b>then </b>
(  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> &amp;  <a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:25"/>
 <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">OU0</font> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K2" target="_self">OSCl</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2">Constants</a> <font color="Maroon">OU0</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a></i>;<br/>


</div>
